#149663

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₂²×C₃₂
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,e^b=ef

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56859);