Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437632 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23);(9,11)(10,15)(12,13)(14,19)(16,23)(17,24)(18,22)(20,21);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,13)(14,20)(15,17)(18,24)(19,21)(22,23);(9,13)(10,17)(11,12)(14,21)(15,16)(18,23)(19,20)(22,24);(11,19)(12,20)(15,22)(16,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318118, #72318139, #72318181, … |
| maximal normal subgroup of | #72318118, #72318139, #72318181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352721);