Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17086 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,29,39)(3,13,16,40,17,8,26,32)(4,18,20,35,21,41,27,9)(5,22,24,42,25,10,28,38)(14,44,46,30,47,51,49,33)(15,43,23,36,48,31,50,58)(19,53,55,37,56,52,57,34)(45,62,54,61,64,63,59,60);(1,3)(2,8)(4,14)(5,23)(6,26)(7,17)(9,33)(10,36)(11,13)(12,40)(15,24)(16,29)(18,51)(19,54)(20,49)(21,47)(22,58)(25,50)(27,46)(28,48)(30,35)(31,42)(32,39)(34,62)(37,63)(38,43)(41,44)(45,55)(52,61)(53,60)(56,59)(57,64);(1,4,7,21)(2,9,11,35)(3,14,17,47)(5,19,25,56)(6,27,29,20)(8,30,13,33)(10,37,22,34)(12,41,39,18)(15,45,48,64)(16,49,26,46)(23,59,50,54)(24,57,28,55)(31,61,43,60)(32,44,40,51)(36,63,58,62)(38,53,42,52);(1,5)(2,10)(3,15)(4,19)(6,28)(7,25)(8,31)(9,34)(11,22)(12,42)(13,43)(14,45)(16,50)(17,48)(18,52)(20,57)(21,56)(23,26)(24,29)(27,55)(30,60)(32,36)(33,61)(35,37)(38,39)(40,58)(41,53)(44,62)(46,59)(47,64)(49,54)(51,63)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383488);