#94166

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
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, prime radical	C₂³	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

131 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 131 conjugacy classes of subgroups — too many to draw. Only its 68 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)(9,10,13,18,15,21,17,32,16,22,29,19,36,38,37,40,12,20,28,39,30,11,23,24,31,26,14,34,33,27,35,25);(2,5)(3,7)(9,11,14,25,30,22,37,39,16,27,13,32,33,20,23,34,12,21,29,40,15,26,35,18,31,38,28,24,36,10,17,19)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1362);