Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
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 | C22×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6369 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)(7,8)(9,10,13,15,14,18,12,17)(11,16,21,22,19,23,20,24);(1,2,3,4)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(6,8)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,24)(18,23);(11,19)(16,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284735, #72284766, #72284953, … |
| maximal normal subgroup of | #72284735, #72284766, #72284953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278622);