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×#3055 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,16,14,11,12,18)(13,17,23,21,22,19,20,24);(5,7,8,6)(9,11,14,10)(12,18,15,16)(13,21,22,24)(17,20,19,23);(2,4)(9,12,14,15)(10,16,11,18)(13,20,22,23)(17,21,19,24);(9,13,12,20,14,22,15,23)(10,17,18,24,11,19,16,21);(13,22)(17,19)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,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^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284234, #72284404 |
| maximal normal subgroup of | #72284234, #72284404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289176);