#8095850

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

4719 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)(18,23)(24,25)(26,30)(31,32);(2,5)(3,7)(10,14)(11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(17,20,22,19)(18,25,23,24)(21,27,29,28)(26,31,30,32);(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)

Polycyclic presentation:

2^9|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, 8003046);