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 | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(19,29)(20,25)(21,37)(22,28)(23,27)(24,33)(26,36)(30,44)(31,40)(32,41)(34,45)(35,43)(38,42)(39,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,33)(18,24,28,29)(20,30,36,48)(21,34,23,32)(25,39,26,44)(27,45,37,41)(31,38,47,35)(40,43,46,42);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,25)(19,30)(21,38)(22,36)(23,35)(24,39)(26,28)(27,43)(29,44)(31,32)(33,48)(34,47)(37,42)(40,41)(45,46);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20)(18,26)(19,31)(21,35)(22,36)(23,38)(24,40)(25,28)(27,43)(29,46)(30,32)(33,47)(34,48)(37,42)(39,41)(44,45);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,27,28,37)(19,32,33,34)(20,35,36,38)(24,41,29,45)(25,42,26,43)(30,47,48,31)(39,46,44,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7438781);