#6347427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347398
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	C₂²×C₄	C₂⁵
prime radical	#60970	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2610 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,15)(14,27)(16,22)(17,26)(18,25)(19,23)(20,24)(21,29)(28,32)(30,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,55)(44,51)(45,50)(46,53)(47,52)(48,54)(49,57)(56,63)(58,61)(59,62)(60,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,21)(10,13)(11,26)(12,22)(18,28)(19,29)(20,30)(23,31)(24,32)(25,27)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,60)(54,61)(59,64)(62,63);(1,4,7,20)(2,9,12,24)(3,14,17,30)(5,19,6,18)(8,21,22,32)(10,25,11,23)(13,27,26,31)(15,29,16,28)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,55)(51,60)(54,62)(58,64)(61,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,18,20,19)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,31,32,27)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(1,5,7,6)(2,11,12,10)(3,16,17,15)(4,19,20,18)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,27,32,31)(33,37,38,48)(34,42,41,54)(35,44,45,58)(36,46,47,59)(39,51,50,61)(40,53,52,62)(43,56,57,64)(49,60,55,63)

Polycyclic presentation:

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

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, 6254623);