Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| 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 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4778 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)(6,7)(9,10,15,18)(11,13,16,14)(12,17,20,24)(19,21,23,22);(5,6)(7,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,23)(22,24);(2,4)(5,7,8,6)(10,16)(11,18)(13,14)(17,23)(19,24)(21,22);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24);(12,20)(17,24)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339107);