#493822

order	512
nilpotency class	4
isoclinism	isoclinic class of #483412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

925 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401018);