Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,18)(7,19,23,43)(8,24,29,39)(9,15,31,42)(10,17,11,16)(13,32,36,22)(14,27,40,28)(20,37,25,34)(21,46,51,61)(26,56,58,53)(30,44,33,41)(35,50,62,49)(38,59,64,57)(45,55,52,48)(47,60,54,63)(65,66,68,67);(1,2)(3,4)(5,7)(6,13)(8,25)(9,21)(10,32)(11,22)(12,23)(14,37)(15,35)(16,43)(17,19)(18,36)(20,29)(24,52)(26,54)(27,48)(28,55)(30,50)(31,51)(33,49)(34,40)(38,63)(39,45)(41,61)(42,62)(44,46)(47,58)(53,68)(56,65)(57,66)(59,67)(60,64);(2,4)(5,8,10,27,12,29,11,28)(6,14,16,39,18,40,17,24)(7,20,22,48,23,25,32,55)(9,26,30,57,31,58,33,59)(13,34,19,45,36,37,43,52)(15,38,41,56,42,64,44,53)(21,47,49,66,51,54,50,67)(35,60,46,65,62,63,61,68);(5,9,12,31)(6,15,18,42)(7,21,23,51)(8,26,29,58)(10,30,11,33)(13,35,36,62)(14,38,40,64)(16,41,17,44)(19,46,43,61)(20,47,25,54)(22,49,32,50)(24,53,39,56)(27,57,28,59)(34,60,37,63)(45,65,52,68)(48,66,55,67)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301793);