#8689736

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

5142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596932);