#8197074

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

4867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104270);