#8460672

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

4863 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:

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

Polycyclic presentation:

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

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, 8367868);