#93938

order	512
nilpotency class	5
isoclinism	isoclinic class of #93766
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1333
commutator	C₂×C₁₆	C₂×C₈
Frattini, cosocle	C₂×#346	C₂²
prime radical	C₂³	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 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)(9,10,12,20,15,22,28,39,13,21,31,37,27,40,30,38,17,24,26,36,14,34,29,19,16,23,32,18,33,35,11,25);(1,3,4,8)(2,5,6,7)(10,18)(11,26)(12,30)(13,16)(14,27)(15,33)(19,22)(20,23)(21,36)(24,37)(25,35)(28,31)(29,32)(34,39)(38,40)

Polycyclic presentation:

2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=ce,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1134);