Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,17)(14,19)(15,21)(20,22)(23,24);(1,2,4,6)(3,8,7,5)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24);(9,12,13,21)(10,15)(11,19,20,24)(14,22)(16,17)(18,23);(11,20)(14,23)(18,22)(19,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72282374, #72283077, #72283086 |
| maximal normal subgroup of | #72282374, #72283077, #72283086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391895);