#495447

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci

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, 402643);