Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395163 |
| 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 | |
1688 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)(7,19,24,36)(8,25,28,42)(9,29,32,15)(10,17,11,16)(13,23,37,22)(14,33,39,27)(20,45,48,34)(21,49,52,44)(26,55,57,54)(30,40,31,41)(35,50,58,51)(38,56,59,60)(43,53,62,47)(46,61,66,64)(63,65,67,68);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,36,18,37,16,19)(8,20,33,53,28,48,27,47)(9,21,31,51,32,52,30,50)(14,34,42,62,39,45,25,43)(15,35,41,49,29,58,40,44)(26,46,60,68,57,66,56,65)(38,61,55,67,59,64,54,63);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,33)(12,28)(13,34)(15,38)(16,25)(17,42)(18,39)(19,43)(21,46)(22,47)(23,53)(24,48)(29,59)(30,56)(31,60)(32,57)(35,61)(36,62)(37,45)(40,54)(41,55)(44,63)(49,67)(50,65)(51,68)(52,66)(58,64);(5,9,10,30,12,32,11,31)(6,15,16,40,18,29,17,41)(7,21,22,50,24,52,23,51)(8,26,27,56,28,57,33,60)(13,35,19,44,37,58,36,49)(14,38,25,54,39,59,42,55)(20,46,47,65,48,66,53,68)(34,61,43,63,45,64,62,67)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302361);