#509117

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#30400
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 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)(3,14)(4,21)(5,27)(6,15)(7,10)(8,13)(9,16)(11,17)(12,28)(18,41)(19,36)(20,35)(22,39)(23,52)(24,57)(25,37)(26,50)(29,34)(30,60)(31,44)(32,45)(33,43)(38,54)(40,47)(42,48)(46,49)(51,61)(53,64)(55,59)(56,62)(58,63);(1,3,8,19,9,20,6,7)(2,10,15,35,16,36,13,14)(4,22,24,17,26,32,25,47)(5,28,30,18,31,33,29,48)(11,37,39,50,40,57,45,21)(12,27,42,34,43,44,41,60)(23,53,54,55,56,46,58,51)(38,49,52,59,63,64,62,61);(1,4)(2,11)(3,17)(5,23)(6,24)(7,32)(8,25)(9,26)(10,21)(12,38)(13,39)(14,37)(15,45)(16,40)(18,46)(19,22)(20,47)(27,59)(28,51)(29,54)(30,58)(31,56)(33,55)(34,49)(35,57)(36,50)(41,52)(42,62)(43,63)(44,61)(48,53)(60,64);(1,5)(2,12)(3,18)(4,23)(6,30)(7,33)(8,29)(9,31)(10,34)(11,38)(13,42)(14,44)(15,41)(16,43)(17,46)(19,28)(20,48)(21,49)(22,51)(24,58)(25,54)(26,56)(27,35)(32,55)(36,60)(37,61)(39,62)(40,63)(45,52)(47,53)(50,64)(57,59)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416313);