Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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×#3284 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,20)(11,21,23,24)(12,19,14,17)(13,22,16,18);(9,11,15,23)(10,16,20,13)(12,22,14,18)(17,21,19,24);(2,4)(9,12,15,14)(10,17,20,19)(11,22,23,18)(13,24,16,21);(9,13)(10,18)(11,17)(12,24)(14,21)(15,16)(19,23)(20,22);(5,7,8,6)(9,12,15,14)(10,17,20,19)(11,22,23,18)(13,24,16,21)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,d^b=df,f^b=fi,f^d=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284404, #72285046, #72285241 |
| maximal normal subgroup of | #72284404, #72285046, #72285241 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291129);