Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3513 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(1,3)(2,5)(4,8)(6,7);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(3,8)(5,7);(2,6)(5,7)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72268383, #72268386, #72268403, … |
| maximal normal subgroup of | #72268383, #72268386, #72268403, … |
| maximal quotient of | #72313958, #72313964, #72314039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300722);