#8152238

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

5171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059434);