#72328136

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 9132 conjugacy classes of subgroups; only the 3101 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=j,e^2=ghij,b^a=bh,c^b=cg,e^a=eghi,e^c=eghi,f^a=fj,f^b=fi,f^d=fi,f^e=fg

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328255
maximal normal subgroup of	#72328255

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.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
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);