#93688

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	C₂×C₄⋊C₁₆	C₂²
socle	C₂²	#1314
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂×C₄⋊C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

159 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 36 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)(3,6,8,5)(4,7)(9,10,11,16,12,17,20,28,14,18,21,29,23,30,33,38,13,15,19,25,22,26,31,35,24,27,32,36,34,37,39,40);(2,5)(3,8)(6,7)(10,15)(17,26)(18,27)(30,37)

Polycyclic presentation:

2^9|a^2=d,c^2=gi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=cgi,d^b=degi,d^c=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 884);