Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,31)(26,35)(28,41)(29,42)(30,43)(32,40)(33,48)(36,45)(37,46)(38,47)(39,44);(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,35,47)(18,23,26,27,25,38,34,43)(20,29,33,46,32,28,44,45)(24,37,39,42,40,36,48,41);(2,5)(6,8)(10,13)(14,16)(17,20,22,32)(18,24,25,40)(19,28,30,29)(21,33,35,44)(23,36,38,37)(26,39,34,48)(27,41,43,42)(31,45,47,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,37)(25,40)(26,39)(27,42)(28,30)(31,46)(34,48)(35,44)(36,38)(41,43)(45,47);(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,32)(26,34)(31,47)(33,44)(36,37)(41,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402797);