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 | |
2813 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)(9,10,15,19)(11,20,21,24)(12,22,23,16)(13,18,14,17);(1,2)(3,4)(6,7)(9,11,13,22,15,21,14,16)(10,12,17,24,19,23,18,20);(5,6,8,7)(9,12,15,23)(10,16,19,22)(11,18,21,17)(13,24,14,20);(2,4)(5,7,8,6)(9,13,15,14)(10,17,19,18)(11,16,21,22)(12,20,23,24);(9,11,13,22,15,21,14,16)(10,12,17,24,19,23,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470480);