#505575

order	512
nilpotency class	4
isoclinism	isoclinic class of #499953
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27582
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412771);