#72336581

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 9095 conjugacy classes of subgroups; only the 2997 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,e^2=hij,f^2=ghi,b^a=bh,c^a=cghi,c^b=cghj,d^a=dhij,d^b=dhij,e^a=eghij,e^b=eghi,e^c=eghi,e^d=ehij,f^a=fj,f^c=fghi,f^d=fgij,f^e=fhj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336654
maximal normal subgroup of	#72336654

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