#6468117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=gh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375313);