Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1552 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)(17,18)(19,28)(20,24)(21,25)(22,27)(23,33)(26,30)(29,43)(31,45)(32,38)(34,40)(35,48)(36,42)(37,44)(39,46)(41,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(19,29)(20,34)(23,38)(24,40)(28,43)(32,33)(36,37)(42,44);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,30,32,22,29,21,33)(18,23,25,43,27,38,26,28)(20,35,36,46,34,31,37,47)(24,41,44,45,40,39,42,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,34)(18,24,27,40)(19,31,29,35)(21,37,30,36)(23,39,38,41)(25,44,26,42)(28,45,43,48)(32,47,33,46);(2,5)(6,8)(10,14)(13,15)(17,21,22,30)(18,25,27,26)(19,32,29,33)(20,36,34,37)(23,28,38,43)(24,42,40,44)(31,46,35,47)(39,48,41,45)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7237495);