Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,28)(20,33)(21,27)(22,24)(23,38)(25,32)(26,34)(29,46)(30,39)(31,44)(35,45)(36,43)(37,42)(40,48)(41,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,34)(25,39)(27,38)(28,44)(29,35)(36,37)(47,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,32,22,31,21,30)(18,25,27,28,24,39,38,44)(20,35,36,48,34,29,37,47)(26,41,43,46,33,40,42,45);(2,6)(5,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,37)(22,34)(23,36)(24,33)(25,40)(27,43)(28,45)(30,48)(31,35)(32,47)(38,42)(39,41)(44,46);(10,14)(13,15)(17,21,22,23)(18,27,24,38)(19,30,31,32)(20,36,34,37)(25,28,39,44)(26,42,33,43)(29,47,35,48)(40,46,41,45)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816542);