Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,12)(9,11)(13,14,20,26)(15,24,29,39)(16,30,34,22)(17,25,19,23)(18,36,28,21)(27,42,31,44)(32,41,35,43)(33,40,37,38);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,19,28,20,29,17,18)(14,21,25,24,26,36,23,39)(16,31,32,37,34,27,35,33)(22,40,41,42,30,38,43,44);(5,8,9,10)(6,7,11,12)(13,16,20,34)(14,22,26,30)(15,27,29,31)(17,35,19,32)(18,37,28,33)(21,38,36,40)(23,43,25,41)(24,44,39,42);(2,4)(6,11)(7,12)(13,17,20,19)(14,23,26,25)(15,18,29,28)(16,32,34,35)(21,39,36,24)(22,41,30,43)(27,33,31,37)(38,44,40,42);(13,18,17,29,20,28,19,15)(14,24,23,21,26,39,25,36)(16,33,35,27,34,37,32,31)(22,42,43,40,30,44,41,38)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=g,d^2=i,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,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449162);