#512929

order	512
nilpotency class	5
isoclinism	isoclinic class of #512929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224992, #72224997, #72225000, …
maximal normal subgroup of	#72224992, #72224997, #72225000, …
maximal quotient of	C₂×#512929

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420125);