#92962

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle, prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

204 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2)(3,4)(6,11,12,15)(7,8)(9,17)(13,20,19,14)(16,18)(21,22,24,28,25,30,33,36,26,31,34,35,32,29,23,27);(2,4)(5,6,8,13,17,20,7,15,10,12,18,19,9,14,16,11)(22,27,29,35,31,36,30,28)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,c^b=chi,d^b=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158);