#501270

order	512
nilpotency class	4
isoclinism	isoclinic class of #501270
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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=h,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=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ehi,f^d=fi,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, 408466);