Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3879 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)(7,19,23,36)(8,14,27,39)(9,28,31,40)(10,17,11,16)(13,24,35,22)(15,29,41,30)(20,45,48,43)(21,49,51,34)(25,54,56,62)(26,42,32,38)(33,53,60,47)(37,57,61,55)(44,50,58,52)(46,59,66,68)(63,65,64,67);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,24)(11,22)(12,23)(14,33)(15,34)(16,36)(17,19)(18,35)(25,46)(26,53)(27,48)(28,58)(29,52)(30,50)(31,51)(32,47)(37,59)(38,43)(39,60)(40,44)(41,49)(42,45)(54,64)(55,67)(56,66)(57,65)(61,68)(62,63);(2,4)(5,8)(6,14)(7,20)(9,25)(10,32)(11,26)(12,27)(13,33)(15,37)(16,42)(17,38)(18,39)(19,43)(21,46)(22,53)(23,48)(24,47)(28,54)(29,57)(30,55)(31,56)(34,59)(35,60)(36,45)(40,62)(41,61)(44,63)(49,68)(50,67)(51,66)(52,65)(58,64);(5,9,12,31)(6,15,18,41)(7,21,23,51)(8,25,27,56)(10,30,11,29)(13,34,35,49)(14,37,39,61)(16,28,17,40)(19,44,36,58)(20,46,48,66)(22,52,24,50)(26,57,32,55)(33,59,60,68)(38,62,42,54)(43,63,45,64)(47,67,53,65)Polycyclic presentation:
2^9|a^2=i,d^2=i,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,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301120);