#72326877

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 9634 conjugacy classes of subgroups; only the 3077 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:

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

Polycyclic presentation:

2^10|d^2=i,b^a=bg,c^b=ch,d^b=di,d^c=dj,e^d=eghi,f^a=fj,f^b=fhj,f^c=fj,f^d=fhj,f^e=fgi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326937
maximal normal subgroup of	#72326937

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