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 | |
4725 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)(9,10,13,18,15,11,14,17)(12,16,21,23,22,19,20,24);(2,4)(5,6)(7,8)(9,10)(11,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23);(6,8)(9,11,15,10)(12,19,22,16)(13,18,14,17)(20,24,21,23);(6,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23);(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24)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,e^a=ei,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, 6345080);