#122311

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

302 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29507);