Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,26)(21,27)(22,28)(23,29)(25,32)(30,39)(31,40)(33,43)(34,41)(35,44)(36,45)(37,42)(38,46)(47,48);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,28,25)(20,33,36,47)(21,31,23,37)(26,43,45,48)(27,40,29,42)(30,38,34,35)(39,46,41,44);(10,14)(11,15)(17,19)(18,25)(20,34)(21,37)(22,32)(23,31)(24,28)(26,39)(27,40)(29,42)(30,36)(33,38)(35,47)(41,45)(43,44)(46,48);(10,14)(13,16)(17,20,21,38,22,36,23,35)(18,26,27,46,28,45,29,44)(19,30,37,47,32,34,31,33)(24,39,42,48,25,41,40,43);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,27,28,29)(19,31,32,37)(20,35,36,38)(24,40,25,42)(26,44,45,46)(30,47,34,33)(39,48,41,43)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518199);