#512991

order	512
nilpotency class	5
isoclinism	isoclinic class of #512991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33607	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 368 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224992, #72225002, #72225003, …
maximal normal subgroup of	#72224992, #72225002, #72225003, …
maximal quotient of	C₂×#512991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420187);