#6352004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2590 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:

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

Polycyclic presentation:

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

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, 6259200);