Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2806 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)(10,15)(12,20)(13,22)(16,23)(17,24)(18,21);(1,2)(3,4)(6,7)(9,10,13,15,14,17,22,24)(11,16,18,23,19,12,21,20);(5,6,8,7)(9,11,14,19)(10,16,17,12)(13,18,22,21)(15,23,24,20);(2,4)(5,7,8,6)(10,17)(11,19)(15,24)(18,21);(9,12,13,20,14,16,22,23)(10,18,15,19,17,21,24,11)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=g,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,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470371);