#9223142

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

4881 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9130338);