#72321832

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ C₂×#2966 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^10|c^2=h,e^2=h,g^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ch,d^c=dhj,e^a=ef,e^b=ehj,e^c=eh,e^d=ehj,g^a=gi,g^b=gi,i^a=ij,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321911
maximal normal subgroup of	#72321911

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