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 | #39335 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1727 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)(17,18,22,24)(19,28,31,41)(20,32,33,26)(21,27,23,38)(25,36,39,30)(29,46,34,45)(35,44,37,43)(40,48,42,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(25,39)(27,38)(28,41)(29,34)(35,37)(47,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,30,22,31,21,36)(18,25,27,28,24,39,38,41)(20,34,35,47,33,29,37,48)(26,42,44,46,32,40,43,45);(2,6)(5,7)(11,16)(13,15)(17,20,22,33)(18,26,24,32)(19,29,31,34)(21,37,23,35)(25,40,39,42)(27,44,38,43)(28,45,41,46)(30,48,36,47);(10,14)(13,15)(17,21,22,23)(18,27,24,38)(19,30,31,36)(20,35,33,37)(25,41,39,28)(26,43,32,44)(29,47,34,48)(40,45,42,46)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,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^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815800);