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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 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)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,45)(31,40)(33,39)(34,48)(35,42)(36,43)(37,44)(38,46)(41,47);(1,3)(2,6)(4,7)(5,8)(17,19,22,32)(18,24,27,29)(20,34,36,47)(21,33,23,31)(25,41,43,48)(26,40,28,39)(30,37,46,35)(38,44,45,42);(9,11,12,16)(10,13,14,15)(17,20,22,36)(18,25,27,43)(19,30,32,46)(21,35,23,37)(24,38,29,45)(26,42,28,44)(31,34,33,47)(39,41,40,48);(2,7)(3,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,40)(25,42,43,44)(30,34,46,47)(38,41,45,48);(1,4,5,6)(2,3,7,8)(18,27)(24,29)(25,43)(26,28)(38,45)(39,40)(41,48)(42,44)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7415765);