#72336579

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 9907 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)(4,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,29)(30,32);(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32);(10,11)(12,16)(26,29)(27,32);(2,3)(6,7)(10,11)(12,16)(19,24)(21,23)(27,32)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,24)(22,23)(25,27,28,32)(26,30,29,31)

Polycyclic presentation:

2^10|a^2=g,e^2=hij,f^2=ghi,b^a=bh,c^a=cghi,c^b=ci,d^a=dhij,e^a=eghij,e^c=eghi,e^d=ehij,f^a=fj,f^c=fghi,f^d=fgi,f^e=fh

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