Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #411625 |
| 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 | |
1820 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)(7,12,19,31)(8,13,22,34)(9,24,27,52)(10,28,25,15)(14,36,37,26)(17,38,40,29)(18,42,44,66)(20,46,43,32)(21,47,49,67)(23,51,48,35)(30,55,56,45)(33,57,58,50)(39,61,63,65)(41,54,62,60)(53,64,59,68);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,20)(11,19)(13,29)(14,30)(15,32)(16,31)(21,39)(22,40)(23,41)(24,42)(25,43)(26,45)(27,44)(28,46)(33,53)(34,38)(35,54)(36,55)(37,56)(47,65)(48,62)(49,63)(50,64)(51,60)(52,66)(57,68)(58,59)(61,67);(2,4)(5,8)(6,13)(7,17)(9,21)(10,23)(11,22)(12,29)(14,33)(15,35)(16,34)(18,39)(19,40)(20,41)(24,47)(25,48)(26,50)(27,49)(28,51)(30,53)(31,38)(32,54)(36,57)(37,58)(42,65)(43,62)(44,63)(45,64)(46,60)(52,67)(55,68)(56,59)(61,66);(5,9,25,36,11,27,10,26)(6,14,28,52,16,37,15,24)(7,18,43,55,19,44,20,45)(8,21,48,57,22,49,23,50)(12,30,46,66,31,56,32,42)(13,33,51,67,34,58,35,47)(17,39,62,68,40,63,41,64)(29,53,60,61,38,59,54,65)Polycyclic presentation:
2^9|a^2=i,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318823);