#512971

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2226
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,14,19,34)(4,20)(5,11,26,12)(6,27,30,53)(7,31,17,9)(10,36)(13,43,45,29)(16,47)(18,41,52,42)(21,54)(22,40)(23,55)(24,58)(25,35,61,62)(28,46)(32,57)(33,60)(37,51)(38,59)(39,63)(44,48)(49,56)(50,64);(1,3,5,17,8,19,26,7)(2,9,12,14,15,31,11,34)(4,16,21,48,24,50,23,32)(6,18,25,43,30,52,61,29)(10,33,38,46,40,39,37,56)(13,35,42,27,45,62,41,53)(20,44,55,47,58,57,54,64)(22,49,59,63,36,28,51,60);(1,4)(2,10)(3,16)(5,21)(6,28)(7,32)(8,24)(9,33)(11,37)(12,38)(13,44)(14,46)(15,40)(17,48)(18,51)(19,50)(20,53)(22,43)(23,26)(25,60)(27,58)(29,36)(30,49)(31,39)(34,56)(35,55)(41,64)(42,47)(45,57)(52,59)(54,62)(61,63);(1,5,8,26)(2,11,15,12)(3,17,19,7)(4,21,24,23)(6,25,30,61)(9,34,31,14)(10,37,40,38)(13,41,45,42)(16,48,50,32)(18,43,52,29)(20,54,58,55)(22,59,36,51)(27,35,53,62)(28,60,49,63)(33,56,39,46)(44,64,57,47)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=i,e^2=i,f^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420167);