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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1509 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,34)(20,24,32,38)(21,25,31,37)(27,33,41,45)(29,35,40,44)(30,36,42,46)(39,43,47,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(20,30)(21,31)(24,36)(25,37)(27,39)(29,40)(32,42)(33,43)(35,44)(38,46)(41,47)(45,48);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,31,42,22,32,21,30)(18,24,37,46,26,38,25,36)(19,27,40,47,28,41,29,39)(23,33,44,48,34,45,35,43);(2,6)(7,8)(19,28)(20,32)(23,34)(24,38)(29,40)(30,42)(35,44)(36,46)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517131);