#501031

order	512
nilpotency class	4
isoclinism	isoclinic class of #501028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

971 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:

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

Polycyclic presentation:

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

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, 408227);