#6372545

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄³	C₂³
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279741);