#6350878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258074);