#72234003

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 9705 conjugacy classes of subgroups; only the 3013 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=i,e^2=ij,c^a=ch,c^b=cgij,d^a=dg,d^b=di,d^c=dj,e^a=eh,e^b=ehi,e^c=eh,e^d=ei,f^c=fhj,f^e=fj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234056
maximal normal subgroup of	#72234056

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