Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,18,10,17)(7,19,24,39,23,47,22,13)(8,25,28,42,30,14,29,48)(9,31,36,46,35,15,34,45)(20,40,26,58,52,49,56,37)(21,50,32,38,55,43,54,65)(27,57,33,44,61,41,60,62)(51,67,53,66,68,64,59,63);(1,2)(3,4)(5,7,10,22,11,23,12,24)(6,13,16,19,18,39,17,47)(8,26,28,52,30,56,29,20)(9,32,34,21,35,54,36,55)(14,40,42,37,25,49,48,58)(15,43,45,65,31,50,46,38)(27,51,33,53,61,68,60,59)(41,63,44,64,57,66,62,67);(2,4)(5,8,12,29,11,30,10,28)(6,14,17,48,18,25,16,42)(7,20,24,56,23,52,22,26)(9,33,36,27,35,60,34,61)(13,37,47,40,39,58,19,49)(15,44,46,41,31,62,45,57)(21,53,32,51,55,59,54,68)(38,64,50,63,65,67,43,66);(5,9,11,35)(6,15,18,31)(7,21,23,55)(8,27,30,61)(10,36,12,34)(13,38,39,65)(14,41,25,57)(16,46,17,45)(19,50,47,43)(20,51,52,68)(22,32,24,54)(26,59,56,53)(28,60,29,33)(37,63,58,66)(40,64,49,67)(42,62,48,44)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ei,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383812);