Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,22,27)(29,37,46,48)(30,38,35,43)(32,40,45,47)(33,41,36,44);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,30)(22,35)(23,31)(25,40)(26,38)(27,43)(28,39)(29,33)(34,45)(36,46)(37,41)(42,47)(44,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,43)(27,38)(28,39)(32,33)(34,46)(36,45)(40,41)(42,48)(44,47);(10,14)(13,16)(17,20,23,34)(18,25,28,42)(19,29,31,46)(21,33,22,36)(24,37,39,48)(26,41,27,44)(30,45,35,32)(38,47,43,40);(1,4,5,3)(2,7,8,6)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,e^2=fi,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,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7436378);