Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,12,23,24,21,16,19,13,17,15,11,20,22);(1,2,3,4)(9,11,16,23)(10,12,19,15)(13,18,14,17)(20,22,24,21);(6,8)(9,10)(11,12)(13,21)(14,22)(15,23)(16,19)(17,24)(18,20);(9,12,16,15)(10,11,19,23)(13,20,14,24)(17,21,18,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72319728, #72319745, #72319777, … |
| maximal normal subgroup of | #72319728, #72319745, #72319777, … |
| maximal quotient of | #72322452, #72322658 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401696);