Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396373 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1294 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)(7,19,23,44)(8,24,28,14)(9,30,33,42)(10,17,11,16)(13,34,37,22)(15,31,43,32)(20,38,47,51)(21,48,50,36)(25,35,54,62)(26,55,57,59)(27,40,29,41)(39,58,52,56)(45,60,64,49)(46,65,66,61)(53,63,68,67);(1,2)(3,4)(5,7)(6,13)(8,25)(9,21)(10,34)(11,22)(12,23)(14,38)(15,36)(16,44)(17,19)(18,37)(20,27)(24,51)(26,53)(28,54)(29,47)(30,45)(31,60)(32,49)(33,50)(35,40)(39,63)(41,62)(42,64)(43,48)(46,56)(52,67)(55,61)(57,68)(58,66)(59,65);(2,4)(5,8,10,27,12,28,11,29)(6,14,16,40,18,24,17,41)(7,20,22,25,23,47,34,54)(9,26,31,56,33,57,32,58)(13,35,19,38,37,62,44,51)(15,39,42,55,43,52,30,59)(21,46,49,53,50,66,60,68)(36,61,45,63,48,65,64,67);(5,9,10,31,12,33,11,32)(6,15,16,42,18,43,17,30)(7,21,34,60,23,50,22,49)(8,26,27,56,28,57,29,58)(13,36,44,64,37,48,19,45)(14,39,40,55,24,52,41,59)(20,46,54,68,47,66,25,53)(35,61,51,67,62,65,38,63)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319278);