#72326822

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 9275 conjugacy classes of subgroups; only the 3057 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)(9,10,12,15)(11,14,16,13)(17,18,21,24)(19,22,23,20)(25,26,28,30)(27,31,32,29);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,26,28,30)(27,29,32,31);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(3,7)(6,8)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,23)(19,24)(21,22)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^10|a^2=j,b^2=h,b^a=bg,c^a=cj,c^b=ch,d^a=dh,d^b=dh,e^a=eghi,e^b=eghi,f^a=fj,f^b=fhij,f^d=fhij,f^e=fhj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326937, #72327044
maximal normal subgroup of	#72326937, #72327044

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