#72329204

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 10148 conjugacy classes of subgroups; only the 3061 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)(3,7)(4,5)(6,8)(17,18)(19,24)(20,22)(21,23)(25,26)(27,30)(28,31)(29,32);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,19,21,22)(18,20,23,24)(25,27)(26,29)(28,32)(30,31);(25,26)(27,30)(28,31)(29,32);(2,4)(6,7)(10,11)(12,16)(19,22)(20,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,24)(20,22)(21,23)(25,26)(27,29)(28,31)(30,32);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(17,20,21,24)(18,22,23,19)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^10|b^2=h,e^2=ghj,f^2=hij,b^a=bi,c^b=cg,d^a=dij,d^b=dh,e^a=eij,e^b=egj,e^d=eghj,f^a=fij,f^b=fhij,f^d=fhij,f^e=fg

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329242
maximal normal subgroup of	#72329242
maximal quotient of	#72331532

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