Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,34,42)(19,23,30,36,29,25,40,41)(20,31,33,45,32,24,37,39)(28,38,44,48,43,35,46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,26)(27,36)(31,38)(32,43)(33,44)(34,40)(37,46)(39,47)(41,42)(45,48);(10,14)(11,15)(17,20,22,33,21,32,34,37)(18,24,27,39,26,31,42,45)(19,28,30,44,29,43,40,46)(23,35,36,47,25,38,41,48);(2,5)(6,8)(10,14)(13,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,38)(27,41)(31,35)(32,43)(33,44)(34,40)(36,42)(37,46)(39,48)(45,47);(1,3,4,7)(2,6,5,8)(19,29)(23,25)(28,43)(30,40)(35,38)(36,41)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319053);