#8658718

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

4574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,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, 8565914);