#501080

order	512
nilpotency class	4
isoclinism	isoclinic class of #501080
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24378
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408276);