#6410981

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di

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, 6318177);