Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,29)(20,34)(21,26)(22,28)(23,27)(24,32)(25,38)(30,39)(31,40)(33,41)(35,42)(36,44)(37,43)(45,46)(47,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,33,21,32)(18,24,26,41,27,40,28,29)(20,35,38,47,37,30,36,46)(25,42,34,45,44,39,43,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,36,23,37,22,38)(18,25,28,43,27,44,26,34)(19,30,32,47,33,35,31,46)(24,39,29,45,40,42,41,48);(10,13)(14,16)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,36,37,38)(24,29,40,41)(25,34,44,43)(30,46,35,47)(39,45,42,48);(9,11,12,15)(10,14,13,16)(18,27)(20,37)(24,40)(26,28)(29,41)(30,35)(36,38)(46,47)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401055);