#8664384

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

4984 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)(9,10)(11,13)(12,15)(14,16)(17,18,22,27)(19,26,29,23)(20,25,21,24)(28,31,30,32);(1,3,6,4)(2,7,5,8)(10,11)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27);(9,10,13,11)(12,14,16,15)(17,20,22,21)(18,24,27,25)(19,28,29,30)(23,31,26,32);(2,5)(7,8)(9,11,13,10)(12,15,16,14)(17,21,22,20)(18,25,27,24)(19,28,29,30)(23,31,26,32);(1,4,6,3)(2,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,29)(28,31)(30,32);(1,5,6,2)(3,8,4,7)(17,19)(18,26)(20,28)(21,30)(22,29)(23,27)(24,32)(25,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 8571580);