#122956

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle, prime radical	C₂²	C₈×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

146 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 512

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

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

Polycyclic presentation:

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

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, 30152);