#6350805

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
prime radical	#62953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2544 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,28)(27,32)(30,31)(33,34,40,46)(35,42,49,56)(36,41,48,57)(37,43,54,62)(38,44,39,45)(47,58,63,59)(50,55,51,64)(52,60,53,61);(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)(33,35,40,49)(34,41,46,57)(36,38,48,39)(37,47,54,63)(42,45,56,44)(43,55,62,64)(50,53,51,52)(58,60,59,61);(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)(34,42)(35,38)(37,51)(39,49)(40,48)(41,45)(43,59)(44,57)(46,56)(47,52)(50,54)(53,63)(55,61)(58,62)(60,64);(4,16)(9,24)(13,27)(17,18)(21,31)(25,29)(26,32)(28,30)(33,37)(34,43)(35,47)(36,50)(38,53)(39,52)(40,54)(41,55)(42,58)(44,61)(45,60)(46,62)(48,51)(49,63)(56,59)(57,64);(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)(33,38,40,39)(34,44,46,45)(35,48,49,36)(37,52,54,53)(41,42,57,56)(43,60,62,61)(47,50,63,51)(55,59,64,58)

Polycyclic presentation:

2^9|a^2=i,b^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, 6258001);