Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,17,13,16,22,24)(11,18,20,21,19,15,23,12);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,17)(13,19)(14,20)(16,18)(21,24)(22,23);(9,12,14,18,13,21,22,15)(10,11,17,20,16,19,24,23);(10,16)(12,21)(15,18)(17,24);(10,16)(11,19)(17,24)(20,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72319459, #72319628, #72319647 |
| maximal normal subgroup of | #72319459, #72319628, #72319647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252890);