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 | |
2681 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)(9,10,15,18)(11,20,17,12)(13,21,19,14)(16,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,11,21,23,15,17,14,22)(10,16,13,12,18,24,19,20);(6,8)(9,12,15,20)(10,17,18,11)(13,22,19,23)(14,16,21,24);(2,4)(6,8)(10,18)(12,20)(13,19)(16,24);(9,13,14,10,15,19,21,18)(11,12,22,16,17,20,23,24)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,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, 7426211);