Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,32)(21,36)(22,25)(23,31)(24,34)(26,30)(28,43)(29,42)(33,41)(35,40)(37,48)(38,46)(39,45)(44,47);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,31,22,29,30,46)(18,23,26,27,25,38,36,42)(20,33,35,48,34,28,44,45)(24,39,40,43,32,37,47,41);(1,3,4,8)(2,5,6,7)(17,20)(18,24)(19,28)(21,35)(22,34)(23,37)(25,32)(26,40)(27,41)(29,33)(30,44)(31,45)(36,47)(38,39)(42,43)(46,48);(2,6)(5,7)(11,15)(13,16)(18,25)(19,29)(20,34)(26,36)(31,46)(35,44)(37,39)(41,43);(17,19,30,46,22,29,21,31)(18,23,36,42,25,38,26,27)(20,28,44,48,34,33,35,45)(24,37,47,43,32,39,40,41)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497802);