#6353288

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2608 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

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, 6260484);