#498802

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1483 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)(3,15)(4,21)(5,26)(6,13)(7,12)(8,14)(9,17)(10,25)(11,29)(16,45)(18,51)(19,39)(20,44)(22,54)(23,38)(24,57)(27,42)(28,41)(30,40)(31,36)(32,61)(33,49)(34,50)(35,47)(37,56)(43,48)(46,64)(52,55)(53,59)(58,62)(60,63);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,35)(11,39)(12,33)(13,42)(14,34)(15,26)(16,25)(17,29)(21,52)(23,48)(24,58)(28,50)(30,49)(31,46)(32,38)(36,62)(37,43)(40,44)(41,51)(45,53)(47,56)(54,57)(55,59)(60,61)(63,64);(1,4)(2,10)(3,16)(5,24)(6,31)(7,25)(8,23)(9,32)(11,37)(12,38)(13,21)(14,36)(15,43)(17,47)(18,22)(19,48)(20,46)(26,53)(27,55)(28,60)(29,59)(30,56)(33,62)(34,52)(35,42)(39,64)(40,63)(41,57)(44,54)(45,51)(49,61)(50,58);(4,23)(10,36)(16,46)(21,38)(22,48)(24,56)(25,31)(32,52)(35,62)(37,57)(43,54)(45,64)(47,58)(53,63)(55,61)(59,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405998);