Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,24,31,39)(20,25,30,38)(21,26,35,43)(22,27,32,40)(29,37,46,48)(33,41,34,42)(36,44,45,47);(1,2)(3,7)(4,8)(5,6)(17,19,23,31)(18,24,28,39)(20,32,30,22)(21,34,35,33)(25,40,38,27)(26,42,43,41)(29,45,46,36)(37,47,48,44);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,22,19,23,30,32,31)(18,25,27,24,28,38,40,39)(21,34,36,46,35,33,45,29)(26,42,44,48,43,41,47,37);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(21,35)(26,43)(29,46)(33,34)(36,45)(37,48)(41,42)(44,47);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,22,36,23,35,32,45)(18,26,27,44,28,43,40,47)(19,29,30,34,31,46,20,33)(24,37,38,42,39,48,25,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751129);