#135818

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8829
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,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,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, 43014);