Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#106 |
| prime radical | C24 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14,15,20)(11,17)(12,19)(16,23,22,21)(18,24);(9,10,11,14,12,16,17,21,13,15,18,20,19,22,24,23);(2,5)(6,8);(10,15)(14,20)(16,22)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225163, #72225318, #72225324, … |
| maximal normal subgroup of | #72225163, #72225318, #72225324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390593);