#122467

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

277 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,28,31,44)(20,24)(21,35)(22,27)(23,30,39,47)(25,33)(26,34)(29,42,32,45)(36,38,46,40)(37,41)(43,48);(17,19,22,31)(18,23,27,39)(20,32,34,29)(21,36,37,46)(24,40,26,38)(25,42,43,45)(28,35,44,41)(30,33,47,48);(17,20,22,34)(18,24,27,26)(19,29,31,32)(21,33,37,48)(23,38,39,40)(25,41,43,35)(28,42,44,45)(30,46,47,36)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dfh,e^b=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29663);