#6356454

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60994	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1474 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,30)(19,23)(21,28)(27,32)(29,31)(33,34)(35,43)(36,41)(37,39)(38,50)(40,46)(42,49)(44,57)(45,56)(47,53)(48,61)(51,52)(54,59)(55,60)(58,64)(62,63);(1,3,6,14,7,15,5,19)(2,8,10,12,11,23,20,22)(4,13,18,28,16,27,17,29)(9,21,25,26,24,31,30,32)(34,39)(40,52)(41,53)(42,50)(43,56)(54,63)(55,61)(57,64);(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)(33,35,37,45)(34,40,39,52)(36,44,47,58)(38,51,49,46)(41,54,53,63)(42,56,50,43)(48,62,60,59)(55,64,61,57);(4,16)(9,24)(13,27)(17,18)(21,31)(25,30)(26,32)(28,29)(33,36)(34,41)(35,44)(37,47)(38,48)(39,53)(40,54)(42,55)(43,57)(45,58)(46,59)(49,60)(50,61)(51,62)(52,63)(56,64);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,16,18)(8,22,23,12)(9,25,24,30)(13,28,27,29)(21,32,31,26)(36,47)(41,53)(44,58)(48,60)(54,63)(55,61)(57,64)(59,62)

Polycyclic presentation:

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

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, 6263650);