#8097779

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

4582 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:

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

Polycyclic presentation:

2^9|c^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004975);