Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461251 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,31,11,17,10,16)(7,18,21,36,22,46,32,13)(8,23,30,39,26,14,25,43)(9,15,33,59,29,42,28,41)(19,47,49,37,50,44,54,34)(20,45,52,63,53,66,60,35)(24,55,58,40,56,38,27,57)(48,62,51,64,67,65,68,61);(5,7,10,21,11,22,12,32)(6,13,31,46,17,36,16,18)(8,19,25,49,26,50,30,54)(9,20,28,52,29,53,33,60)(14,37,43,47,23,34,39,44)(15,35,59,66,42,63,41,45)(24,48,27,51,56,67,58,68)(38,64,57,62,55,61,40,65);(5,8,11,26)(6,14,17,23)(7,19,22,50)(9,27,29,58)(10,30,12,25)(13,34,36,37)(15,40,42,57)(16,43,31,39)(18,44,46,47)(20,51,53,68)(21,54,32,49)(24,33,56,28)(35,62,63,65)(38,59,55,41)(45,61,66,64)(48,60,67,52);(2,4)(5,9,12,33,11,29,10,28)(6,15,31,59,17,42,16,41)(7,20,32,60,22,53,21,52)(8,24,30,58,26,56,25,27)(13,35,46,66,36,63,18,45)(14,38,43,57,23,55,39,40)(19,48,54,68,50,67,49,51)(34,61,44,65,37,64,47,62)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=i,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368494);