Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,31,11,16,10,17)(7,13,29,40,20,36,21,35)(8,22,30,14,24,41,25,38)(9,26,32,39,28,15,27,56)(18,42,49,33,45,50,44,60)(19,46,57,61,47,34,48,62)(23,52,58,55,53,51,54,37)(43,64,68,67,66,63,65,59);(1,2)(3,4)(5,7,10,29,11,20,12,21)(6,13,31,35,16,36,17,40)(8,18,30,44,24,45,25,49)(9,19,27,57,28,47,32,48)(14,33,22,50,38,60,41,42)(15,34,56,61,26,46,39,62)(23,43,58,65,53,66,54,68)(37,59,51,64,55,67,52,63);(2,4)(5,8)(6,14)(7,18)(9,23)(10,30)(11,24)(12,25)(13,33)(15,37)(16,38)(17,41)(19,43)(20,45)(21,49)(22,31)(26,55)(27,58)(28,53)(29,44)(32,54)(34,59)(35,50)(36,60)(39,52)(40,42)(46,67)(47,66)(48,68)(51,56)(57,65)(61,64)(62,63);(5,9,10,27,11,28,12,32)(6,15,31,56,16,26,17,39)(7,19,29,57,20,47,21,48)(8,23,30,58,24,53,25,54)(13,34,35,61,36,46,40,62)(14,37,22,51,38,55,41,52)(18,43,44,65,45,66,49,68)(33,59,50,64,60,67,42,63)Polycyclic presentation:
2^9|a^2=fi,b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308261);