Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1450 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)(3,7)(4,8)(5,6)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,32)(29,43)(30,44)(31,39)(33,47)(34,41)(35,42)(36,48)(38,46)(40,45);(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,39,37,44)(20,33,36,45,34,29,35,46)(25,40,42,47,41,38,48,43);(2,6)(7,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,34)(24,38)(26,42)(27,41)(28,43)(30,33)(31,45)(32,46)(37,48)(39,40)(44,47);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,34)(24,39)(25,41)(28,44)(31,32)(35,36)(42,48);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,35,34,36)(24,28,39,44)(25,42,41,48)(29,45,33,46)(38,43,40,47)Polycyclic presentation:
2^9|b^2=f,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6444036);