#499786

order	512
nilpotency class	4
isoclinism	isoclinic class of #499785
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 ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1238 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406982);