#149204

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

626 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=ei,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56400);