#499654

order	512
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#493
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23798
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283827, #72283857, #72283942, …
maximal normal subgroup of	#72283827, #72283857, #72283942, …
maximal quotient of	C₂×#499654

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406850);