#122310

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
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×#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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 29506);