#8664936

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

4890 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572132);