#94122

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₄×C₃₂
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,12,18,14,20,27,35,15,11,17,22,30,24,33,38,16,21,13,19,26,34,28,36,31,25,29,23,32,37,40,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,23,14,24,28,39,15,21,29,18,30,34,40,35,16,25,12,22,26,37,27,38,31,10,17,19,32,20,33,36)

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,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1318);