Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35634 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,17,19,15)(14,21,22,16)(18,23,24,20);(5,7,8,6)(9,11,13,19)(10,15,12,17)(14,23,22,20)(16,24,21,18);(6,7)(9,12,13,10)(11,18)(14,21,22,16)(15,23)(17,20)(19,24);(2,4)(6,7)(10,12)(11,19)(16,21)(20,23)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330426);