Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1475 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,36)(22,27)(23,26)(24,32)(29,40)(30,41)(31,38)(33,39)(34,44)(35,43)(37,42)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19,22,32)(18,24,27,28)(20,33,35,30)(21,31,23,29)(25,41,43,39)(26,38,36,40)(34,46,37,45)(42,47,44,48);(10,14)(13,16)(19,29)(21,23)(24,38)(26,36)(28,40)(30,45)(31,32)(33,46)(34,37)(39,47)(41,48)(42,44);(2,7)(3,8)(10,14)(13,16)(17,20,22,35)(18,25,27,43)(19,30,32,33)(21,37,23,34)(24,39,28,41)(26,44,36,42)(29,45,31,46)(38,47,40,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,27,36)(19,31,32,29)(20,34,35,37)(24,38,28,40)(25,42,43,44)(30,45,33,46)(39,48,41,47)Polycyclic presentation:
2^9|b^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413100);