Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,28,30,24)(20,31,34,47)(21,36,37,43)(23,38,33,27)(25,40,42,35)(29,45,46,41)(32,44,48,39);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,32,34,48)(21,23,37,33)(25,41,42,45)(27,43,38,36)(29,40,46,35)(31,39,47,44);(10,13)(14,15)(17,20,33,40,22,34,23,35)(18,25,38,47,26,42,27,31)(19,29,37,48,30,46,21,32)(24,39,43,45,28,44,36,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,23,30,22,37,33,19)(18,24,27,36,26,28,38,43)(20,32,35,46,34,48,40,29)(25,39,31,41,42,44,47,45);(1,4,5,3)(2,7,6,8)(18,26)(20,34)(24,28)(27,38)(29,46)(32,48)(35,40)(36,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fgi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505520);