#6366858

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	#62551	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1464 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)(3,11)(4,8)(5,19)(6,9)(7,15)(10,22)(12,20)(13,29)(14,27)(16,30)(17,28)(18,25)(21,26)(23,32)(24,31)(33,34,39,42)(35,40,46,54)(36,47,49,58)(37,43,38,51)(41,52,57,48)(44,59,62,64)(45,55,50,56)(53,60,63,61);(1,3)(2,7)(4,16)(5,20)(6,14)(8,24)(9,12)(10,27)(11,19)(13,17)(15,22)(18,32)(21,31)(23,26)(25,29)(28,30)(33,35,39,46)(34,40,42,54)(36,44,49,62)(37,50,38,45)(41,53,57,63)(43,56,51,55)(47,59,58,64)(48,61,52,60);(1,2)(3,12)(4,8)(5,10)(6,9)(7,14)(11,20)(13,30)(15,27)(16,29)(17,26)(18,25)(19,22)(21,28)(23,31)(24,32)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,51)(50,61)(54,63)(55,59)(56,64);(1,4,6,18)(2,8,9,25)(3,13,14,31)(5,21,22,17)(7,23,12,30)(10,28,19,26)(11,29,27,24)(15,32,20,16)(34,42)(40,54)(41,57)(43,51)(47,58)(53,63)(55,56)(59,64);(2,9)(3,14)(8,25)(10,19)(13,31)(15,20)(16,32)(26,28)(33,37,39,38)(34,43,42,51)(35,45,46,50)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,62,61)(53,59,63,64)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=i,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, 6274054);