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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2906 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)(10,16)(11,20)(12,14)(13,21)(17,24)(18,19);(1,2)(3,4)(6,7)(9,10,14,16,15,19,12,18)(11,21,22,24,20,17,23,13);(5,6,8,7)(9,11,15,20)(10,17,19,21)(12,23,14,22)(13,18,24,16);(2,4)(5,7,8,6)(9,12,15,14)(10,18,19,16)(11,22,20,23)(13,21,24,17);(9,13,12,17,15,24,14,21)(10,11,18,23,19,20,16,22)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,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^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308202, #72308890 |
| maximal normal subgroup of | #72308202, #72308890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470425);