#8096182

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

4726 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:

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

Polycyclic presentation:

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

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, 8003378);