#124227

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

346 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31423);