Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1369 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,34)(21,40)(22,28)(23,29)(24,30)(26,37)(27,32)(31,43)(33,42)(35,41)(36,48)(38,45)(39,44)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,25)(20,35)(21,24)(22,32)(23,33)(26,43)(27,28)(29,42)(30,40)(31,37)(34,41)(36,38)(39,46)(44,47)(45,48);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,23,38,24,39,22,37)(18,26,28,44,30,45,29,34)(19,31,32,46,21,36,33,35)(25,41,42,48,40,47,27,43);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,21,22,32,24,19,23,33)(18,27,29,40,30,42,28,25)(20,36,37,46,39,31,38,35)(26,43,34,47,45,48,44,41);(2,8)(6,7)(17,22,24,23)(18,28,30,29)(19,32,21,33)(20,37,39,38)(25,27,40,42)(26,44,45,34)(31,46,36,35)(41,43,47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fh,d^2=fg,e^2=i,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,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510750);