#72227960

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

18038 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=hi,b^a=bi,c^a=chj,c^b=cg,d^a=dgj,d^b=dj,d^c=dgh,e^a=eghj,e^b=egh,e^c=eghi,e^d=eg,f^a=fj,f^b=fghj,f^c=fghi,f^d=fhi,f^e=fgij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227961
Fitting subgroup of	#72227961
maximal subgroup of	#72227961
maximal normal subgroup of	#72227961

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