#72336400

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 9202 conjugacy classes of subgroups; only the 3017 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,22)(23,24)(26,29)(30,31);(9,11)(10,13)(12,16)(14,15)(25,26)(27,30)(28,29)(31,32);(2,3)(6,7)(10,12)(11,15)(18,19)(22,23)(25,26)(27,30)(28,29)(31,32);(2,3)(6,7)(10,12)(13,16)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,29)(30,32);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,16,15,13)(17,19,21,18)(20,23,24,22)(25,26)(27,31)(28,29)(30,32);(1,4,5,8)(2,6,3,7)(9,13,14,16)(10,15,12,11)(17,20,21,24)(18,22,19,23)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^10|d^2=h,e^2=gij,f^2=gij,b^a=bg,c^a=cj,c^b=ci,d^a=dj,d^c=dh,e^a=ej,e^c=egij,e^d=eghij,f^d=fgh,f^e=fhij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336494
maximal normal subgroup of	#72336494

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