Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350909 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62956 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,26)(4,9,18,24)(5,11,6,10)(8,20,22,15)(14,27,30,31)(17,25,19,23)(21,28,32,29)(33,34,39,44)(35,45,49,55)(36,50,52,42)(37,40,38,43)(41,47,58,48)(46,62,51,61)(53,60,54,56)(57,64,59,63);(1,3,7,16)(2,8,12,22)(4,14,18,30)(5,15,6,20)(9,21,24,32)(10,13,11,26)(17,28,19,29)(23,31,25,27)(34,40)(37,38)(41,55)(42,56)(43,44)(45,58)(47,48)(50,60)(53,54)(57,62)(59,61)(63,64);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,23)(12,24)(13,27)(15,29)(16,30)(20,28)(22,32)(26,31)(33,35,39,49)(34,41,44,58)(36,51,52,46)(37,48,38,47)(40,55,43,45)(42,59,50,57)(53,63,54,64)(56,61,60,62);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,18,19)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,31,32,27)(33,36,39,52)(34,42,44,50)(35,46,49,51)(37,53,38,54)(40,56,43,60)(41,57,58,59)(45,61,55,62)(47,63,48,64);(36,52)(42,50)(46,51)(53,54)(56,60)(57,59)(61,62)(63,64)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258260);