Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2851 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)(9,10,15,19,16,11,14,18)(12,17,23,24,21,20,22,13);(5,7,8,6)(9,11,16,10)(12,20,21,17)(13,23,24,22)(14,18,15,19);(1,2,3,4)(9,12)(10,17)(11,20)(13,18)(14,22)(15,23)(16,21)(19,24);(9,13,14,20,16,24,15,17)(10,12,18,22,11,21,19,23);(12,21)(13,24)(17,20)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284404, #72284589, #72284953 |
| maximal normal subgroup of | #72284404, #72284589, #72284953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289993);