#9228936

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5289 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280721
maximal normal subgroup of	#72280721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9136132);