#501083

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

1027 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,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, 408279);