Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,14)(12,17)(13,21)(15,19)(16,24)(18,20)(22,23);(5,7,8,6)(9,11,20,19,15,18,14,10)(12,17,13,22,24,16,23,21);(1,2,3,4)(9,12,14,23,15,24,20,13)(10,16,11,21,19,17,18,22);(9,13,20,24,15,23,14,12)(10,17,11,22,19,16,18,21)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fgi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224689, #72226904, #72226917, … |
| maximal normal subgroup of | #72224689, #72226904, #72226917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276360);