Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3212 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)(9,10,16,20)(11,21,12,24)(13,18,17,23)(14,19,15,22);(5,6)(7,8)(9,11,15,18,16,12,14,23)(10,17,19,21,20,13,22,24);(2,4)(6,8)(9,12,15,23,16,11,14,18)(10,13,19,24,20,17,22,21);(6,8)(9,13)(10,18)(11,22)(12,19)(14,21)(15,24)(16,17)(20,23);(9,14,16,15)(10,19,20,22)(11,23,12,18)(13,24,17,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346176);