Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
1336 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)(5,6,10,17,12,16,11,34)(7,18,21,42,23,13,22,37)(8,24,27,40,29,43,28,14)(9,30,31,60,33,15,32,41)(19,46,48,35,49,38,25,44)(20,36,50,62,52,45,51,64)(26,55,56,39,58,53,57,59)(47,63,67,65,68,66,54,61);(5,7,12,23)(6,13,16,18)(8,25,29,48)(9,20,33,52)(10,22,11,21)(14,38,40,46)(15,36,30,45)(17,42,34,37)(19,27,49,28)(24,35,43,44)(26,54,58,67)(31,51,32,50)(39,63,59,66)(41,64,60,62)(47,56,68,57)(53,61,55,65);(2,4)(5,8,12,29)(6,14,16,40)(7,19,23,49)(9,26,33,58)(10,28,11,27)(13,35,18,44)(15,39,30,59)(17,43,34,24)(20,47,52,68)(21,25,22,48)(31,57,32,56)(36,61,45,65)(37,46,42,38)(41,55,60,53)(50,54,51,67)(62,66,64,63);(5,9)(6,15,16,30)(7,20)(8,26)(10,31)(11,32)(12,33)(13,36,18,45)(14,39,40,59)(17,41,34,60)(19,47)(21,50)(22,51)(23,52)(24,53,43,55)(25,54)(27,56)(28,57)(29,58)(35,61,44,65)(37,62,42,64)(38,63,46,66)(48,67)(49,68)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303312);