Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,14,18,16,12,15,17)(11,19,20,23,22,24,21,13);(2,4)(5,6)(7,8)(9,11,16,22)(10,13,12,23)(14,21,15,20)(17,19,18,24);(6,8)(9,12,16,10)(11,13,22,23)(14,18,15,17)(19,20,24,21);(6,8)(9,13)(10,11)(12,22)(14,19)(15,24)(16,23)(17,21)(18,20);(9,14,16,15)(10,17,12,18)(11,20,22,21)(13,24,23,19)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345089);