Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,14)(12,13)(15,16)(18,21)(19,22)(23,24);(5,6,7,8)(10,13)(14,16);(2,4)(17,18,22,24,20,23,19,21);(9,11,12,15)(10,14,13,16)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgi,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72249728, #72249752, #72249763, … |
| maximal normal subgroup of | #72249728, #72249752, #72249763, … |
| maximal quotient of | #72250822, #72322765 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261362);