#512993

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	#33611	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

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

normal subgroups only

This group has 358 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:

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

Polycyclic presentation:

2^9|c^2=eh,d^2=i,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, #72225001, #72225004, …
maximal normal subgroup of	#72224992, #72225001, #72225004, …
maximal quotient of	C₂×#512993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420189);