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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,33)(21,39)(22,27)(23,29)(25,37)(26,32)(28,35)(30,42)(31,41)(34,40)(36,45)(38,43)(44,46)(47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,29,26)(20,34,38,36)(21,22,31,35)(25,42,44,48)(27,41,28,39)(30,46,47,37)(33,40,43,45);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,35,46,23,38,22,37)(18,25,27,43,29,44,28,33)(19,30,31,36,32,47,21,34)(24,40,39,48,26,45,41,42);(9,11,12,16)(10,14,13,15)(17,21,22,32,23,31,35,19)(18,26,28,39,29,24,27,41)(20,36,37,30,38,34,46,47)(25,40,33,42,44,45,43,48);(1,4,5,3)(2,7,6,8)(17,22,23,35)(18,27,29,28)(19,31,32,21)(20,37,38,46)(24,41,26,39)(25,43,44,33)(30,36,47,34)(40,42,45,48)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=g,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504623);