Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,12,19,14,17,13,16)(11,20,21,24,22,15,23,18);(10,15)(11,21)(12,13)(16,18)(17,20)(19,24)(22,23);(1,3,4,8,5,2,6,7)(10,16,17,19)(11,22)(15,18,20,24)(21,23);(10,17)(15,20)(16,19)(18,24)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216353, #72216441, #72216443, … |
| maximal normal subgroup of | #72216353, #72216441, #72216443, … |
| maximal quotient of | #72216334, C4×#12067, C2×#483590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390786);