Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14022 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1025 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)(3,14)(4,20,25,37)(5,26)(6,30)(7,12)(8,13)(9,17)(10,23,38,24)(11,27)(15,31,43,49)(16,44)(18,34)(19,33)(21,35,48,40)(22,36,54,60)(28,39)(29,56)(32,47)(41,45)(42,61,62,59)(46,58)(50,53,55,52)(51,57,64,63);(1,3)(2,9,13,34)(4,21)(5,16)(6,17)(7,18)(8,19)(10,35,38,40)(11,32,39,44)(12,14,30,33)(15,24)(20,49,37,31)(22,51)(23,43)(25,48)(26,41,56,58)(27,45)(28,46)(29,47)(36,59,60,61)(42,53)(50,63,55,57)(52,62)(54,64);(1,4,7,24,8,25,6,23)(2,10,12,37,13,38,30,20)(3,15,17,21,19,43,18,48)(5,22,28,53,29,54,27,52)(9,31,33,35,34,49,14,40)(11,36,26,55,39,60,56,50)(16,42,45,51,47,62,46,64)(32,57,58,59,44,63,41,61);(1,5,6,27,8,29,7,28)(2,11,12,26,13,39,30,56)(3,16,17,45,19,47,18,46)(4,22,23,52,25,54,24,53)(9,32,14,41,34,44,33,58)(10,36,37,55,38,60,20,50)(15,42,21,51,43,62,48,64)(31,57,40,61,49,63,35,59)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275863);