#493460

order	512
nilpotency class	4
isoclinism	isoclinic class of #493458
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2131 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)(3,13)(4,18)(5,10)(6,25,26,11)(7,12)(8,16,29,43)(9,21)(14,38)(15,36)(17,37)(19,45)(20,31,50,46)(22,51)(23,53,54,33)(24,34)(27,55,44,40)(28,42,32,52)(30,49)(35,41)(39,60)(47,59)(48,58,64,62)(56,63,57,61);(1,3,7,17)(2,8,12,29)(4,14,21,41)(5,22,24,15)(6,16,26,43)(9,27,18,44)(10,32,34,28)(11,13,25,37)(19,47,49,39)(20,40,50,55)(23,52,54,42)(30,57,45,56)(31,35,46,38)(33,51,53,36)(48,63,64,61)(58,60,62,59);(1,4)(2,9,12,18)(3,14)(5,19)(6,20)(7,21)(8,27,29,44)(10,30,34,45)(11,31,25,46)(13,35,37,38)(15,39)(16,40)(17,41)(22,47)(23,48)(24,49)(26,50)(28,56,32,57)(33,58,53,62)(36,59,51,60)(42,61)(43,55)(52,63)(54,64);(1,5,7,24)(2,10,12,34)(3,15,17,22)(4,19,21,49)(6,23,26,54)(8,28,29,32)(9,30,18,45)(11,33,25,53)(13,36,37,51)(14,39,41,47)(16,42,43,52)(20,48,50,64)(27,56,44,57)(31,58,46,62)(35,59,38,60)(40,61,55,63)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^e=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400656);