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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1510 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,20,25)(21,26,36,45)(22,37,24,27)(30,43,32,39)(31,40,33,41)(34,48,47,42)(35,46,38,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,25,28,29)(21,34,36,47)(22,31,24,33)(26,42,45,48)(27,40,37,41)(30,38,32,35)(39,44,43,46);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,29)(21,30,36,32)(22,31,24,33)(26,43,45,39)(27,41,37,40)(34,38,47,35)(42,46,48,44);(10,14)(13,16)(17,21,22,38,23,36,24,35)(18,26,37,44,28,45,27,46)(19,30,33,47,20,32,31,34)(25,39,40,48,29,43,41,42);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,37)(19,31,20,33)(21,35,36,38)(25,40,29,41)(26,44,45,46)(30,47,32,34)(39,42,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7519640);