Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(17,18,23,27)(19,28,30,40)(20,32,33,25)(21,26,22,37)(24,36,39,31)(29,43,45,42)(34,47,35,41)(38,44,46,48);(9,10)(11,14)(12,13)(15,16)(17,19,21,36,23,30,22,31)(18,24,37,40,27,39,26,28)(20,29,34,48,33,45,35,44)(25,38,47,43,32,46,41,42);(2,6)(7,8)(10,13)(14,16)(17,20,23,33)(18,25,27,32)(19,29,30,45)(21,34,22,35)(24,38,39,46)(26,41,37,47)(28,42,40,43)(31,44,36,48);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(24,39)(25,32)(28,40)(31,36)(34,35)(41,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,36)(20,34,33,35)(24,40,39,28)(25,41,32,47)(29,44,45,48)(38,43,46,42)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454087);