Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(19,31)(20,35)(21,27)(22,28)(23,29)(24,30)(25,33)(26,34)(32,43)(36,48)(37,41)(38,44)(39,45)(40,46)(42,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33,24,34,22,20)(18,25,28,31,30,35,29,26)(21,32,38,37,40,47,39,36)(27,41,45,43,46,48,44,42);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,34,24,33,23,19)(18,26,29,35,30,31,28,25)(21,37,39,32,40,36,38,47)(27,43,44,41,46,42,45,48);(9,11,12,15)(10,14,13,16)(17,21,24,40)(18,27,30,46)(19,32,34,47)(20,36,33,37)(22,39,23,38)(25,41,35,48)(26,42,31,43)(28,45,29,44);(10,13)(14,16)(17,22,24,23)(18,28,30,29)(19,20,34,33)(21,38,40,39)(25,31,35,26)(27,44,46,45)(32,37,47,36)(41,42,48,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426760);