Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420377 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15277 |
| 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 | |
1281 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)(3,15,19,39)(4,10)(5,24,27,41)(6,12)(7,14)(8,13)(9,18,33,28)(11,26,38,30)(16,43,20,42)(17,32)(21,50,53,55)(22,35)(23,36)(25,37)(29,40)(31,46,58,45)(34,54,62,52)(44,57)(47,60)(48,59)(49,64)(51,63)(56,61);(1,3)(2,9)(4,20)(5,25)(6,28)(7,18)(8,19)(10,31)(11,37)(12,39)(13,33)(14,15)(16,23)(17,26)(21,51)(22,45)(24,32)(27,47)(29,46)(30,48)(34,61)(35,43)(36,58)(38,60)(40,42)(41,59)(44,54)(49,52)(50,64)(53,56)(55,57)(62,63);(1,4,6,22,8,23,7,29)(2,10,12,35,13,36,14,40)(3,16,18,45,19,20,28,46)(5,21,26,52,27,53,30,54)(9,31,15,42,33,58,39,43)(11,34,24,55,38,62,41,50)(17,44,25,56,48,49,47,51)(32,57,37,61,59,64,60,63);(1,5,6,26,8,27,7,30)(2,11,12,24,13,38,14,41)(3,17,18,25,19,48,28,47)(4,21,22,52,23,53,29,54)(9,32,15,37,33,59,39,60)(10,34,35,55,36,62,40,50)(16,44,45,56,20,49,46,51)(31,57,42,61,58,64,43,63)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327573);