#499777

order	512
nilpotency class	4
isoclinism	isoclinic class of #499777
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)(3,16)(4,23)(5,27)(6,14)(7,13)(8,15)(9,19)(10,25)(11,24)(12,31)(17,51)(18,35)(20,59)(21,45)(22,50)(26,41)(28,47)(29,44)(30,39)(32,46)(33,42)(34,62)(36,57)(37,58)(38,52)(40,60)(43,61)(48,53)(49,55)(54,64)(56,63);(1,3)(2,9)(4,18)(5,28)(6,20)(7,21)(8,22)(10,38)(11,35)(12,45)(13,36)(14,47)(15,37)(16,27)(17,25)(19,31)(23,49)(24,56)(26,53)(29,55)(30,58)(32,57)(33,54)(34,41)(39,59)(40,48)(42,63)(43,64)(44,51)(46,50)(52,60)(61,62);(1,4)(2,10)(3,17)(5,24)(6,33)(7,25)(8,26)(9,34)(11,39)(12,40)(13,41)(14,23)(15,42)(16,48)(18,20)(19,52)(21,53)(22,54)(27,44)(28,55)(29,31)(30,61)(32,60)(35,50)(36,63)(37,49)(38,47)(43,46)(45,64)(51,59)(56,58)(57,62);(1,4)(2,11)(3,18)(5,29)(6,25)(7,33)(8,26)(9,35)(10,39)(12,42)(13,43)(14,44)(15,40)(16,49)(17,20)(19,56)(21,54)(22,53)(23,27)(24,31)(28,55)(30,60)(32,61)(34,50)(36,64)(37,48)(38,59)(41,46)(45,63)(47,51)(52,58)(57,62)

Polycyclic presentation:

2^9|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,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,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, 406973);