#509855

order	512
nilpotency class	4
isoclinism	isoclinic class of #509855
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31064
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=efh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defh,g^a=gi,c^b=cg,d^b=de,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417051);