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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 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)(9,10,12,13)(11,15,16,14)(17,18)(19,25)(20,34)(21,40)(22,28)(23,29)(24,30)(26,37)(27,33)(31,43)(32,42)(35,41)(36,46)(38,45)(39,44)(47,48);(2,6)(7,8)(10,13)(11,16)(17,19)(18,25)(20,35)(21,23)(22,32)(24,33)(26,43)(27,30)(28,42)(29,40)(31,37)(34,41)(36,39)(38,47)(44,46)(45,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,23,38,24,39,22,37)(18,26,28,44,30,45,29,34)(19,31,32,36,33,47,21,35)(25,41,40,48,27,46,42,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,22,33,24,32,23,19)(18,27,29,40,30,25,28,42)(20,36,37,31,39,35,38,47)(26,41,34,43,45,46,44,48);(2,6)(7,8)(10,13)(14,15)(17,22,24,23)(18,28,30,29)(19,32,33,21)(20,37,39,38)(25,42,27,40)(26,44,45,34)(31,36,47,35)(41,43,46,48)Polycyclic presentation:
2^9|a^2=gh,c^2=fh,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510614);