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 | |
1564 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)(9,10,11,16,19,23,24,21,13,15,20,22,12,17,18,14);(1,2,3,4)(10,14)(11,18)(12,19)(15,21)(16,17)(20,24)(22,23);(6,8)(10,14)(11,18)(12,19)(15,21)(16,17)(20,24)(22,23);(10,15)(14,21)(16,22)(17,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,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 | #72240843, #72250110, #72250142, … |
| maximal normal subgroup of | #72240843, #72250110, #72250142, … |
| maximal quotient of | #72264918, #72279467, #72286629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401694);