#496020

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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:

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

Polycyclic presentation:

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

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, 403216);