#501252

order	512
nilpotency class	4
isoclinism	isoclinic class of #501252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24424
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

1035 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 408448);