#512841

order	512
nilpotency class	5
isoclinism	isoclinic class of #512841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
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

754 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 144 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)(4,17)(5,20)(6,24)(9,30)(10,28)(11,35)(12,36)(14,39)(15,41)(16,38)(18,23)(19,45)(21,54)(22,48)(25,53)(26,29)(27,31)(32,61)(33,58)(34,55)(37,51)(40,43)(42,64)(44,59)(46,47)(49,57)(50,56)(52,63)(60,62);(1,2)(3,9)(4,18)(5,23)(6,26)(7,10)(8,17)(11,20)(12,24)(13,31)(14,40)(15,43)(16,44)(19,49)(21,55)(22,50)(25,56)(27,45)(28,48)(29,54)(30,39)(32,41)(33,38)(34,53)(35,46)(36,47)(37,57)(42,62)(51,60)(52,61)(58,63)(59,64);(1,3)(2,9)(4,19)(5,15)(6,16)(7,13)(8,27)(10,31)(11,32)(12,33)(14,22)(17,45)(18,49)(20,41)(21,52)(23,43)(24,38)(25,37)(26,44)(28,30)(29,58)(34,60)(35,62)(36,59)(39,48)(40,50)(42,46)(47,64)(51,53)(54,63)(55,61)(56,57);(3,13)(9,31)(14,19)(15,37)(16,38)(27,30)(32,60)(33,44)(39,45)(40,49)(41,51)(42,52)(43,57)(58,59)(61,62)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224992, #72224993, #72224996, …
maximal normal subgroup of	#72224992, #72224993, #72224996, …
maximal quotient of	C₂×#512841

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420037);