#94124

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂³	C₄×C₁₆
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 186 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=g,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, 1320);