#93937

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	#360
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

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

normal subgroups only

This group has 128 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,e^2=i,f^2=hi,g^2=h,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, 1133);