Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #23853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,22,24)(19,28,31,44)(20,33,36,26)(21,27,23,38)(25,30,39,32)(29,46,34,45)(35,43,37,42)(40,47,41,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(25,39)(26,33)(27,38)(28,44)(40,41)(42,43)(45,46);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,31,23,32)(18,25,38,44,24,39,27,28)(20,34,37,47,36,29,35,48)(26,41,42,45,33,40,43,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,36)(18,26,24,33)(19,29,31,34)(21,37,23,35)(25,40,39,41)(27,43,38,42)(28,45,44,46)(30,48,32,47);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,24,38)(19,30,31,32)(20,35,36,37)(25,28,39,44)(26,42,33,43)(29,47,34,48)(40,46,41,45)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6816858);