#124470

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂³	C₂×#104
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fgh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31666);