Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,19)(12,21)(13,16)(14,22)(15,18)(17,24)(20,23);(5,6)(7,8)(9,11,14,16,15,20,24,21)(10,13,17,19,18,12,22,23);(1,2,3,4)(6,8)(9,12,15,13)(10,16,18,21)(11,17,20,22)(14,23,24,19);(5,6)(7,8)(9,13,15,12)(10,16,18,21)(11,17,20,22)(14,19,24,23);(6,8)(11,20)(12,13)(16,21)(19,23)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490335);