Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,44)(35,43)(45,47)(46,48);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,31)(21,28)(22,30)(24,40)(25,38)(26,37)(27,32)(29,35)(33,46)(34,45)(36,42)(39,43)(41,48)(44,47);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,22,33)(18,24,25,41)(19,27,30,45)(21,34,35,32)(23,36,38,47)(26,44,43,42)(28,31,29,46)(37,40,39,48);(1,4,5,3)(2,6,8,7)(19,28)(21,35)(23,37)(26,43)(27,31)(29,30)(32,34)(36,40)(38,39)(42,44)(45,46)(47,48);(18,25)(23,38)(24,41)(26,43)(36,47)(37,39)(40,48)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349126);