Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| 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×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2762 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)(11,20)(12,23)(13,18)(14,21)(15,19)(16,17)(22,24);(9,11,14,17,15,22,13,23)(10,16,18,20,19,12,21,24);(2,4)(9,12)(10,17)(11,21)(13,20)(14,24)(15,16)(18,22)(19,23);(5,7,8,6)(9,13,15,14)(10,18,19,21)(11,17,22,23)(12,20,16,24);(11,22)(12,16)(17,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284404, #72284669, #72285158 |
| maximal normal subgroup of | #72284404, #72284669, #72285158 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280077);