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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10,15,17)(11,19,21,24)(12,22,23,16)(13,20,18,14);(1,2)(3,4)(5,6)(7,8)(9,11,20,22,15,21,14,16)(10,12,13,24,17,23,18,19);(6,8)(9,12,15,23)(10,16,17,22)(11,18,21,13)(14,19,20,24);(2,4)(6,8)(10,17)(12,23)(13,18)(19,24);(9,13,14,10,15,18,20,17)(11,19,16,23,21,24,22,12)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,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=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426243);