#8697578

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

4763 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^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=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, 8604774);