#94123

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
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	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 189 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=g,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, 1319);