#94165

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	#346
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

134 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1361);