#496377

order	512
nilpotency class	4
isoclinism	isoclinic class of #496375
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21810
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60649	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1043 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)(3,8,17,27)(4,10,19,34)(5,13)(6,9)(7,24)(11,23)(12,14)(15,39,36,28)(16,26)(18,29,44,53)(20,30,45,55)(21,47,37,35)(22,32)(25,48,50,41)(31,38)(33,57,56,46)(40,59)(42,52)(43,54)(49,60)(51,58)(61,64)(62,63);(1,3,14,36,5,17,7,15)(2,8,12,28,13,27,24,39)(4,16,37,58,19,42,21,40)(6,18,38,48,22,44,23,41)(9,29,31,50,32,53,11,25)(10,26,35,51,34,52,47,59)(20,43,57,62,45,61,46,60)(30,54,56,63,55,64,33,49);(2,9)(3,15)(6,23)(7,14)(8,25)(10,30)(11,12)(13,32)(16,40)(17,36)(18,44)(20,46)(21,37)(22,38)(24,31)(26,49)(27,50)(28,53)(29,39)(33,35)(34,55)(42,58)(43,61)(45,57)(47,56)(51,64)(52,63)(54,59);(1,4,5,19)(2,10,13,34)(3,16,17,42)(6,20,22,45)(7,21,14,37)(8,26,27,52)(9,30,32,55)(11,33,31,56)(12,35,24,47)(15,40,36,58)(18,43,44,61)(23,46,38,57)(25,49,50,63)(28,51,39,59)(29,54,53,64)(41,60,48,62)

Polycyclic presentation:

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

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, 403573);