#72334228

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

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 8565 conjugacy classes of subgroups; only the 3037 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:

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

Polycyclic presentation:

2^10|c^2=i,d^2=hi,e^2=hij,b^a=bj,c^b=cg,d^b=dj,d^c=dh,e^b=ej,e^c=ehj,e^d=ej,f^a=fi,f^b=fhij,f^c=fi,f^d=fgj,f^e=fghi

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334244
maximal normal subgroup of	#72334244

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