Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1761 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,37)(28,34)(29,38)(30,41)(31,40)(32,35)(33,36)(39,45)(42,48)(43,46)(44,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(20,30)(22,33)(24,34)(25,39)(27,42)(29,43)(31,44)(32,45)(35,41)(37,47)(40,48);(10,14)(11,15)(17,19)(18,24)(20,31)(21,28)(22,29)(23,34)(25,40)(26,38)(27,32)(30,44)(33,43)(35,37)(36,46)(39,48)(41,47)(42,45);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,30,21,32,33,45)(18,25,26,41,23,35,36,39)(19,27,29,42,28,31,43,44)(24,37,38,48,34,40,46,47);(2,7)(6,8)(18,23)(19,28)(25,35)(26,36)(27,31)(29,43)(39,41)(42,44)Polycyclic presentation:
2^9|b^2=fi,d^2=fhi,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7516769);