#93314

order	512
nilpotency class	4
isoclinism	isoclinic class of #93313
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1281
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	#1726	C₂²
socle	C₂	#7018
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₂×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

166 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 510);