#499781

order	512
nilpotency class	4
isoclinism	isoclinic class of #499781
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1261 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406977);