#93441

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#329
commutator	C₄²	C₄×C₈
Frattini, cosocle	#1726	C₂²
socle	C₂	#7030
prime radical	C₂³	#344
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,12,8,14,29,43,9,15,30,13,7,33,6,31)(3,16,18,46,22,39,53,64,20,47,52,63,21,56,19,54)(4,11,24,40,27,41,25,58,28,42,48,44,26,59,32,45)(10,34,35,51,38,49,62,50,37,61,36,23,57,55,60,17);(1,3,6,19,7,21,30,52,9,20,29,53,8,22,5,18)(2,10,13,36,14,38,31,60,15,37,12,35,33,57,43,62)(4,23,25,34,26,50,24,55,28,51,32,61,27,17,48,49)(11,39,40,64,41,47,58,63,42,56,44,54,59,16,45,46)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=g,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 637);