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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,14,19,16,21,15,20)(11,18,22,24,23,13,12,17);(2,4)(5,6)(7,8)(9,11)(10,17)(12,14)(13,19)(15,22)(16,23)(18,20)(21,24);(6,8)(9,12,16,22)(10,18,21,13)(11,14,23,15)(17,20,24,19);(6,8)(9,13)(10,12)(11,19)(14,17)(15,24)(16,18)(20,23)(21,22);(9,14,16,15)(10,19,21,20)(11,12,23,22)(13,24,18,17)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6345226);