Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1490 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,28,43)(20,31,32,25)(22,36,35,26)(24,38,37,30)(29,44,33,42)(34,41,45,47)(39,46,40,48);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,28)(20,32)(26,36)(30,38)(34,45)(39,40)(42,44);(10,14)(11,15)(17,19,22,30,21,28,35,38)(18,24,26,27,23,37,36,43)(20,33,34,48,32,29,45,46)(25,40,41,44,31,39,47,42);(2,5)(6,8)(17,20)(18,25)(19,29)(21,32)(22,34)(23,31)(24,39)(26,41)(27,42)(28,33)(30,46)(35,45)(36,47)(37,40)(38,48)(43,44);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(19,28)(20,32)(24,37)(25,31)(27,43)(30,38)(34,45)(41,47)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6671797);