Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #44932 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,16,18)(11,13,14,15)(12,17,22,24)(19,23,21,20);(2,4)(6,7)(9,11,15,10,16,14,13,18)(12,19,20,17,22,21,23,24);(5,6,8,7)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,22)(18,24);(1,2,3,4)(5,7,8,6)(9,13,16,15)(10,14,18,11)(12,20,22,23)(17,19,24,21);(9,14,15,18,16,11,13,10)(12,21,20,24,22,19,23,17)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=f,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521270);