#135830

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#481
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8830
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fi,g^b=gi,d^c=di

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, 43026);