#72340344

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

2790 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=ij,c^2=i,b^a=bj,c^a=cfg,c^b=cgh,d^a=dhj,d^b=dfghi,d^c=dfg,e^a=ehj,e^b=egh,e^c=eghi,f^c=fi,f^d=fj,f^e=fj,g^a=gi,g^b=gij,g^c=gi,g^d=gj,h^a=hj,h^b=hij,h^c=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340368, #72340561
maximal normal subgroup of	#72340368, #72340561

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