#6366769

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1458 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273965);