#8096123

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

5061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,7,8)(3,4,6,5)(9,10,13,14)(11,15,16,12)(18,20)(22,23);(1,3,7,6)(2,5,8,4)(11,16)(12,15)(17,18,21,20)(19,23,24,22);(1,4,7,5)(2,3,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,21,24)(18,22,20,23);(1,4,7,5)(2,3,8,6)(9,12)(10,11)(13,15)(14,16)(17,20,21,18)(19,23,24,22);(1,5,7,4)(2,3,8,6);(1,6,7,3)(2,5,8,4)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^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, 8003319);