Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,16,12,22)(11,18,21,24,20,15,19,23);(1,3)(2,5)(4,7)(6,8)(11,19,20,21)(15,23,18,24);(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,18)(17,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266586, #72266644, #72267258 |
| maximal normal subgroup of | #72266586, #72266644, #72267258 |
| maximal quotient of | #72266589 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23543);