Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,22,28)(19,29,32,24)(20,25,34,42)(21,26,23,27)(30,43,44,37)(31,38,36,39)(33,40,35,41)(45,47,46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,28,29)(20,30,34,44)(21,31,23,36)(25,37,42,43)(26,39,27,38)(33,45,35,46)(40,48,41,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,34)(18,25,28,42)(19,30,32,44)(21,35,23,33)(24,37,29,43)(26,41,27,40)(31,46,36,45)(38,48,39,47);(10,14)(13,15)(17,21)(18,26)(20,33)(22,23)(25,40)(27,28)(30,44)(31,36)(34,35)(37,43)(38,39)(41,42);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,26)(19,31,32,36)(20,33,34,35)(24,38,29,39)(25,41,42,40)(30,45,44,46)(37,47,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7489624);