Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,32)(30,38)(31,45)(33,40)(34,48)(35,43)(36,41)(37,44)(39,47)(42,46);(2,5)(6,8)(17,19,22,32)(18,24,25,29)(20,34,36,46)(21,33,23,30)(26,42,41,48)(27,38,28,40)(31,35,47,37)(39,43,45,44);(9,11,12,16)(10,13,14,15)(18,25)(19,30)(21,23)(24,38)(26,41)(29,40)(31,46)(32,33)(34,47)(35,37)(39,42)(45,48);(17,20,22,36)(18,26,25,41)(19,31,32,47)(21,35,23,37)(24,39,29,45)(27,43,28,44)(30,46,33,34)(38,42,40,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,25,28)(19,30,32,33)(20,35,36,37)(24,40,29,38)(26,43,41,44)(31,46,47,34)(39,48,45,42)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353800);