Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394199 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1574 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)(7,19,24,35)(8,25,29,38)(9,15,31,41)(10,17,11,16)(13,23,36,22)(14,27,39,28)(20,33,48,59)(21,44,50,61)(26,54,56,53)(30,42,32,40)(34,51,60,49)(37,57,62,55)(43,46,52,47)(45,64,66,58)(63,67,68,65);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,35,18,36,16,19)(8,20,28,47,29,48,27,46)(9,21,32,51,31,50,30,49)(14,33,25,52,39,59,38,43)(15,34,42,61,41,60,40,44)(26,45,57,67,56,66,55,65)(37,58,53,68,62,64,54,63);(2,4)(5,8,10,27,12,29,11,28)(6,14,16,38,18,39,17,25)(7,20,22,46,24,48,23,47)(9,26,30,55,31,56,32,57)(13,33,19,43,36,59,35,52)(15,37,40,54,41,62,42,53)(21,45,49,65,50,66,51,67)(34,58,44,63,60,64,61,68);(5,9)(6,15)(7,21)(8,26)(10,30)(11,32)(12,31)(13,34)(14,37)(16,40)(17,42)(18,41)(19,44)(20,45)(22,49)(23,51)(24,50)(25,53)(27,55)(28,57)(29,56)(33,58)(35,61)(36,60)(38,54)(39,62)(43,63)(46,65)(47,67)(48,66)(52,68)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^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,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, 301951);