Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,14,18,13,17,12,16)(11,15,21,24,20,23,19,22);(6,7)(9,10,12,16,13,17,14,18)(11,15,19,22,20,23,21,24);(1,2,3,4)(9,11)(10,15)(12,19)(13,20)(14,21)(16,22)(17,23)(18,24);(5,7,8,6)(9,12,13,14)(10,16,17,18)(11,19,20,21)(15,22,23,24);(11,20)(15,23)(19,21)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^b=dfi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308898, #72308910, #72308922, … |
| maximal normal subgroup of | #72308898, #72308910, #72308922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289933);