Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| 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 | |
1302 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)(5,6,12,18)(7,19,23,44)(8,24,28,14)(9,30,33,42)(10,17,11,16)(13,34,37,22)(15,31,43,32)(20,47,25,45)(21,50,52,36)(26,56,58,60)(27,40,29,41)(35,49,38,55)(39,59,53,57)(46,61,64,51)(48,62,54,63)(65,67,66,68);(5,7)(6,13)(8,25)(9,21)(10,34)(11,22)(12,23)(14,38)(15,36)(16,44)(17,19)(18,37)(20,28)(24,35)(26,54)(27,49)(29,55)(30,46)(31,61)(32,51)(33,52)(39,63)(40,45)(41,47)(42,64)(43,50)(48,58)(53,62)(56,65)(57,67)(59,68)(60,66);(2,4)(5,8,10,27,12,28,11,29)(6,14,16,40,18,24,17,41)(7,20,22,49,23,25,34,55)(9,26,31,57,33,58,32,59)(13,35,19,45,37,38,44,47)(15,39,42,56,43,53,30,60)(21,48,51,67,52,54,61,68)(36,62,46,65,50,63,64,66);(5,9,10,31,12,33,11,32)(6,15,16,42,18,43,17,30)(7,21,34,61,23,52,22,51)(8,26,27,57,28,58,29,59)(13,36,44,64,37,50,19,46)(14,39,40,56,24,53,41,60)(20,48,55,68,25,54,49,67)(35,62,47,66,38,63,45,65)Polycyclic presentation:
2^9|a^2=i,c^2=e,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^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319267);