Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15314 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9)(4,18,20,36)(5,22)(6,12)(7,27)(8,13)(10,21,35,26)(11,23)(14,39,41,48)(15,44)(16,31)(17,32)(19,50,52,34)(24,37)(25,33)(28,38)(29,42,56,43)(30,46)(40,55,51,59)(45,57)(47,58)(49,53,60,54)(61,63,64,62);(1,3)(2,9)(4,14,20,41)(5,15)(6,25)(7,16)(8,17)(10,29,35,56)(11,30)(12,33)(13,32)(18,48,36,39)(19,51,52,40)(21,43,26,42)(22,44)(23,46)(24,47)(27,31)(28,45)(34,59,50,55)(37,58)(38,57)(49,61,60,64)(53,62,54,63);(1,4,6,26,8,20,7,21)(2,10,12,36,13,35,27,18)(3,14,16,43,17,41,25,42)(5,19,23,54,24,52,28,53)(9,29,31,48,32,56,33,39)(11,34,37,60,38,50,22,49)(15,40,45,63,47,51,46,62)(30,55,44,64,57,59,58,61);(1,5,8,24)(2,11,13,38)(3,15,17,47)(4,19,20,52)(6,23,7,28)(9,30,32,57)(10,34,35,50)(12,37,27,22)(14,40,41,51)(16,45,25,46)(18,49,36,60)(21,53,26,54)(29,55,56,59)(31,44,33,58)(39,61,48,64)(42,62,43,63)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330616);