#72285907

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(2,7)(3,4)(5,13)(10,12)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,37)(29,42)(31,35)(32,38)(33,39)(34,41)(36,44)(40,46)(43,48)(45,47);(1,2)(3,8)(4,11)(5,14,13,9)(6,7)(10,16,12,15)(17,19)(18,23)(20,31)(21,29,34,44)(22,30)(24,37)(25,36,41,42)(26,27)(28,33)(32,43,46,45)(35,39)(38,47,40,48);(1,3,6,4)(2,8,7,11)(5,10)(9,16)(12,13)(14,15)(17,20)(18,24)(19,28)(21,32,34,46)(22,33)(23,35)(25,40,41,38)(26,39)(27,37)(29,43,44,45)(30,31)(36,48,42,47);(1,4,6,3)(2,8)(5,12)(7,11)(9,15,14,16)(10,13)(17,21)(18,25,26,41)(19,29)(20,32,33,46)(22,34)(23,36,27,42)(24,38)(28,43,31,45)(30,44)(35,47)(37,48)(39,40);(1,5,6,13)(2,9,7,14)(3,10,4,12)(8,15,11,16)(21,34)(23,27)(24,39)(25,41)(28,31)(35,37)(36,42)(38,40)

Polycyclic presentation:

2^10|b^2=f,c^2=fj,d^2=gj,e^2=j,b^a=bij,c^a=cfj,c^b=cfi,d^a=dfij,d^b=dh,d^c=dfg,e^a=ehij,e^b=efg,e^c=efghij,e^d=ehj,f^d=fi,f^e=fj,g^a=gi,g^b=gj,g^c=gi,g^e=gj,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044, #72340246, #72340432, …
maximal normal subgroup of	#72286044, #72340246, #72340432, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);