Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1472 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)(9,10,12,14)(11,13,15,16)(17,18,23,28)(19,24,20,25)(21,26,35,44)(22,27,37,46)(29,38,31,40)(30,39,32,41)(33,42,47,48)(34,43,36,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,28,25)(21,33,35,47)(22,32,37,30)(26,42,44,48)(27,41,46,39)(29,36,31,34)(38,45,40,43);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,24)(21,31,35,29)(22,32,37,30)(26,40,44,38)(27,41,46,39)(33,34,47,36)(42,43,48,45);(10,14)(13,16)(17,21,23,35)(18,26,28,44)(19,29,20,31)(22,36,37,34)(24,38,25,40)(27,45,46,43)(30,47,32,33)(39,48,41,42);(1,4,5,3)(2,7,6,8)(19,20)(24,25)(29,31)(30,32)(33,47)(38,40)(39,41)(42,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7435877);