#509204

order	512
nilpotency class	4
isoclinism	isoclinic class of #509203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30499
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^d=fh,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416400);