Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(2,6)(3,4)(7,8)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(2,6)(3,4)(7,8);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268383, #72268386, #72268405, … |
| maximal normal subgroup of | #72268383, #72268386, #72268405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105570);