#6351868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#62825	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259064);