#8688256

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

5157 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gi,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, 8595452);