#94859

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄³	C₂³
commutator, socle, prime radical	C₂³	C₄³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bgi,c^a=ch,c^b=cg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2055);