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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1469 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,35)(29,42)(30,40)(31,41)(33,39)(34,45)(36,44)(37,46)(38,47)(43,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,27,40)(20,33,35,29)(21,36,38,48)(23,34,37,31)(25,42,32,39)(26,44,47,43)(28,45,46,41);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,35)(18,25,27,32)(19,29,30,33)(21,31,38,34)(23,36,37,48)(24,39,40,42)(26,45,47,41)(28,43,46,44);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,37,33,22,38,23,29)(18,26,46,39,27,47,28,42)(19,20,34,48,30,35,31,36)(24,32,45,43,40,25,41,44);(2,6)(7,8)(20,35)(21,38)(25,32)(26,47)(29,33)(36,48)(39,42)(43,44)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=fhi,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, 7517511);