#72232640

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9683 conjugacy classes of subgroups; only the 3025 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(2,6)(4,7)(10,14)(12,16)(18,22)(20,23)(26,29)(30,31);(1,2,5,6)(3,4,8,7)(9,10,13,14)(11,12,15,16)(17,18,21,22)(19,20,24,23)(26,29)(30,31);(1,3)(2,7)(4,6)(5,8)(9,11,13,15)(10,12,14,16)(25,26)(27,30)(28,29)(31,32);(2,6)(4,7)(10,14)(11,15)(17,19)(18,23)(20,22)(21,24)(25,26,28,29)(27,31,32,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4)(2,8)(3,6)(5,7)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,22)(21,23)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^10|b^2=g,c^2=h,d^2=j,b^a=bg,c^a=cj,c^b=ci,d^a=dj,d^b=dgj,d^c=dhj,e^a=ehj,e^b=ehj,e^d=ehj,f^a=fg,f^d=fi,f^e=fg

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232676
maximal normal subgroup of	#72232676

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