#94036

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

182 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,16,22,18,24,31,39,19,15,21,26,34,28,37,42,20,25,17,23,30,38,32,40,35,29,33,27,36,41,44,43);(2,4)(5,6,8,11,9,12,7,10)(13,15,17,27,18,28,32,43,19,25,33,22,34,38,44,39,20,29,16,26,30,41,31,42,35,14,21,23,36,24,37,40)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1232);