#6356455

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	#537	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,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, 6263651);