Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(17,18,23,28)(19,24,31,42)(20,26,34,44)(21,27,22,37)(25,36,45,33)(29,39,32,46)(30,40,35,41)(38,47,43,48);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,42)(20,32,34,29)(21,35,22,30)(25,43,45,38)(26,46,44,39)(27,41,37,40)(33,47,36,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,28,45)(19,29,31,32)(21,33,22,36)(24,38,42,43)(26,27,44,37)(30,47,35,48)(39,41,46,40);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20)(18,26)(19,29)(21,36)(22,33)(23,34)(24,39)(25,37)(27,45)(28,44)(30,48)(31,32)(35,47)(38,40)(41,43)(42,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,28,37)(19,30,31,35)(20,33,34,36)(24,40,42,41)(25,44,45,26)(29,47,32,48)(38,39,43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7214701);