#147089

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ehi,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54285);