#122959

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 66 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 66 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)(19,27)(20,31)(21,36)(22,24)(23,28)(25,34)(26,33)(29,47)(30,44)(32,42)(35,43)(37,41)(38,46)(39,48)(40,45);(17,19,22,30)(18,23,24,38)(20,32,35,48)(21,37,33,29)(25,41,40,47)(26,42,36,39)(27,43,44,31)(28,45,46,34);(1,3,4,11,5,9,10,16,6,12,13,15,14,2,7,8)(18,24)(23,38)(25,40)(26,36)(27,44)(31,43)(39,42)(41,47)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=e,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=chi,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, 30155);