#122355

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 401 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=dhi,e^b=efi,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, 29551);