#93433

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 86 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)(4,6)(5,8)(9,10,12,19,15,20,28,34,16,21,29,35,14,30,13,18)(11,22,23,17,26,31,38,32,27,37,39,33,25,40,24,36);(1,3)(2,5,6,7)(4,8)(9,11,13,24,14,25,29,39,16,27,28,38,15,26,12,23)(10,17,19,31,20,32,34,37,21,33,35,40,30,36,18,22)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=h,e^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=cg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 629);