#509784

order	512
nilpotency class	4
isoclinism	isoclinic class of #509783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31000
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

657 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416980);