Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #411585 |
| 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 | |
1800 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)(7,19,23,41)(8,14,26,38)(9,27,30,39)(10,17,11,16)(13,31,35,22)(15,28,40,29)(20,45,48,43)(21,50,52,34)(24,53,55,62)(25,42,32,37)(33,49,59,47)(36,57,60,54)(44,56,61,51)(46,58,66,68)(63,67,64,65);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,31)(11,22)(12,23)(14,33)(15,34)(16,41)(17,19)(18,35)(24,46)(25,49)(26,48)(27,44)(28,56)(29,51)(30,52)(32,47)(36,58)(37,43)(38,59)(39,61)(40,50)(42,45)(53,63)(54,67)(55,66)(57,65)(60,68)(62,64);(2,4)(5,8)(6,14)(7,20)(9,24)(10,32)(11,25)(12,26)(13,33)(15,36)(16,42)(17,37)(18,38)(19,43)(21,46)(22,49)(23,48)(27,53)(28,57)(29,54)(30,55)(31,47)(34,58)(35,59)(39,62)(40,60)(41,45)(44,63)(50,68)(51,67)(52,66)(56,65)(61,64);(5,9,10,28,12,30,11,29)(6,15,16,39,18,40,17,27)(7,21,31,56,23,52,22,51)(8,24,32,57,26,55,25,54)(13,34,41,61,35,50,19,44)(14,36,42,62,38,60,37,53)(20,46,47,65,48,66,49,67)(33,58,45,64,59,68,43,63)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318920);