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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(19,24)(20,31)(21,26)(22,27)(23,29)(25,33)(28,38)(30,41)(32,42)(34,45)(35,44)(36,39)(37,43)(40,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,32)(21,29)(22,36)(24,26)(25,42)(27,41)(28,34)(30,35)(31,48)(33,47)(37,46)(38,43)(39,44)(40,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,40)(27,41)(31,38)(32,34)(33,46)(35,36)(37,47)(39,44)(42,45)(43,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,21,33)(18,25,26,31)(19,28,29,46)(22,37,35,34)(23,38,24,40)(27,45,44,43)(30,47,36,32)(39,48,41,42);(1,4,5,3)(2,7,8,6)(18,26)(19,29)(25,31)(27,44)(28,46)(30,36)(32,47)(43,45)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=f,e^2=fi,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, 7438788);