#6351729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351720
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	C₂×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2583 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)(4,18,20,26)(5,11,6,10)(8,15,24,16)(9,21,25,19)(14,22,30,32)(27,29,31,28)(33,34,39,42)(35,45,48,57)(36,41,50,55)(37,44,38,43)(40,51,54,47)(46,60,63,59)(49,58,52,56)(53,61,64,62);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28)(20,30)(21,29)(25,32)(26,27)(33,35,39,48)(34,40,42,54)(36,46,50,63)(37,47,38,51)(41,53,55,64)(43,45,44,57)(49,61,52,62)(56,59,58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,26)(12,25)(13,27)(15,28)(16,29)(17,30)(23,31)(24,32)(33,36)(34,41)(35,46)(37,49)(38,52)(39,50)(40,53)(42,55)(43,56)(44,58)(45,59)(47,61)(48,63)(51,62)(54,64)(57,60);(34,42)(40,54)(41,55)(43,44)(45,57)(53,64)(56,58)(59,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,18,25,26)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,43,42,44)(35,47,48,51)(36,49,50,52)(40,45,54,57)(41,56,55,58)(46,61,63,62)(53,59,64,60)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,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, 6258925);