#6349967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349901
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2603 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,27)(28,31)(30,32)(33,34)(35,45)(36,50)(37,44)(38,43)(39,42)(40,48)(41,52)(46,60)(47,55)(49,56)(51,57)(53,58)(54,63)(59,61)(62,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,28)(18,30)(24,31)(25,32)(26,29)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,54)(42,55)(43,56)(44,45)(50,64)(51,61)(52,62)(53,63)(57,60)(58,59);(1,4,7,17)(2,9,11,25)(3,13,15,28)(5,16,6,18)(8,21,23,32)(10,24,19,29)(12,26,22,31)(14,27,20,30)(33,36,39,53)(34,41,42,57)(35,46,49,63)(37,52,38,51)(40,54,55,60)(43,50,44,58)(45,59,56,64)(47,62,48,61);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,29)(13,27,28,30)(21,31,32,26)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64);(2,11)(4,17)(8,23)(10,19)(12,22)(13,28)(16,18)(27,30)(33,37,39,38)(34,43,42,44)(35,47,49,48)(36,51,53,52)(40,56,55,45)(41,58,57,50)(46,61,63,62)(54,59,60,64)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,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,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306654, #72306657, #72306723
maximal normal subgroup of	#72306654, #72306657, #72306723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257163);