#8687581

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

5131 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)(3,4)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,29)(21,25)(22,23)(24,26)(28,30)(31,32);(1,2,6,5)(3,8,4,7)(11,15)(13,16)(17,19,24,23)(18,22,26,27)(20,28,21,31)(25,30,29,32);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,20,24,21)(18,25,26,29)(19,28,23,31)(22,30,27,32);(1,4,6,3)(2,8,5,7)(9,11,12,15)(10,13,14,16)(17,21,24,20)(18,25,26,29)(19,28,23,31)(22,32,27,30);(1,5,6,2)(3,8,4,7)(9,10)(11,13)(12,14)(15,16)(17,22,24,27)(18,19,26,23)(20,30,21,32)(25,28,29,31);(17,23,24,19)(18,22,26,27)(20,31,21,28)(25,30,29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8594777);