#512843

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

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

normal subgroups only

This group has 744 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:

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

Polycyclic presentation:

2^9|d^2=i,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, #72224994, #72224995, …
maximal normal subgroup of	#72224992, #72224994, #72224995, …
maximal quotient of	C₂×#512843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420039);