#499783

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:

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

Polycyclic presentation:

2^9|a^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, 406979);