#72340237

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

2787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=ghi,e^2=j,b^a=bij,c^a=cf,c^b=cfg,d^a=di,d^b=dfi,d^c=di,e^a=ehij,e^b=ehij,e^d=eghj,f^b=fi,f^c=fi,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gi,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340248, #72340307, #72340309, …
maximal normal subgroup of	#72340248, #72340307, #72340309, …

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.8);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
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);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, 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.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);