#499782

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

1245 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 406978);