Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456505 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,34)(7,18,22,37,23,47,21,13)(8,24,27,40,28,43,26,14)(9,29,32,44,33,15,31,42)(19,35,50,61,51,38,49,45)(20,46,54,36,55,63,53,59)(25,41,57,65,58,56,30,39)(48,60,67,62,68,64,52,66);(1,2)(3,4)(5,7)(6,13,17,37)(8,19)(9,20)(10,21)(11,22)(12,23)(14,38,40,35)(15,36,29,59)(16,18,34,47)(24,45,43,61)(25,48)(26,49)(27,50)(28,51)(30,52)(31,53)(32,54)(33,55)(39,64,65,60)(41,66,56,62)(42,63,44,46)(57,67)(58,68);(2,4)(5,8,12,28)(6,14,17,40)(7,19,23,51)(9,30,33,57)(10,27,11,26)(13,35,37,38)(15,41,29,56)(16,43,34,24)(18,45,47,61)(20,52,55,67)(21,50,22,49)(25,32,58,31)(36,62,59,66)(39,44,65,42)(46,64,63,60)(48,54,68,53);(5,9,12,33)(6,15,17,29)(7,20,23,55)(8,25,28,58)(10,32,11,31)(13,36,37,59)(14,39,40,65)(16,44,34,42)(18,46,47,63)(19,48,51,68)(21,54,22,53)(24,56,43,41)(26,57,27,30)(35,60,38,64)(45,62,61,66)(49,67,50,52)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=i,e^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363845);