#6357608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1785 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)(3,11)(4,9)(5,17)(6,8)(7,13)(10,18)(12,24)(14,23)(15,21)(16,29)(19,27)(20,26)(22,28)(25,31)(30,32)(33,34)(35,40)(36,48)(37,42)(38,52)(39,43)(41,50)(44,57)(45,54)(46,55)(47,61)(49,56)(51,62)(53,60)(58,64)(59,63);(1,3,5,13,6,14,18,27)(2,7,10,11,8,19,17,23)(4,12,16,26,15,25,28,32)(9,20,22,24,21,30,29,31)(33,35)(34,40)(36,44)(37,46)(38,47)(39,45)(41,53)(42,55)(43,54)(48,57)(49,59)(50,60)(51,58)(52,61)(56,63)(62,64);(2,8)(7,19)(9,21)(10,17)(11,23)(20,30)(22,29)(24,31)(33,36,37,49,39,51,38,50)(34,41,52,62,43,56,42,48)(35,44,46,59,45,58,47,60)(40,53,61,64,54,63,55,57);(1,4)(2,9)(3,12)(5,16)(6,15)(7,20)(8,21)(10,22)(11,24)(13,26)(14,25)(17,29)(18,28)(19,30)(23,31)(27,32)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(4,15)(9,21)(12,25)(16,28)(20,30)(22,29)(24,31)(26,32)(33,37,39,38)(34,42,43,52)(35,46,45,47)(36,49,51,50)(40,55,54,61)(41,48,56,62)(44,59,58,60)(53,57,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264804);