Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,13)(14,19)(16,21)(17,23)(18,24)(20,22);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,23)(19,22)(21,24);(1,4,5,3)(2,7,6,8)(9,13)(10,17,18,21)(11,12)(14,22)(15,23,24,16)(19,20)Polycyclic presentation:
2^9|b^2=e,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225671, #72225682, #72225683, … |
| maximal normal subgroup of | #72225671, #72225682, #72225683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355523);