#6349890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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	#62827	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257086);