#72334199

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 9225 conjugacy classes of subgroups; only the 3053 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=ij,e^2=gij,b^a=bi,c^a=cij,c^b=cg,d^b=dghij,d^c=dghij,e^a=eij,e^b=eghij,e^c=eg,e^d=eghij,f^a=fij,f^b=fhj,f^d=fgi,f^e=fghij

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