Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1992 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)(5,6)(7,8)(9,10,13,15)(11,16,12,14)(17,18,21,23)(19,22,24,20);(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,20,23,22);(2,4)(18,19)(20,22)(23,24);(6,8)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237126, #72237135, #72237141, … |
| maximal normal subgroup of | #72237126, #72237135, #72237141, … |
| maximal quotient of | #72250017, #72250040, #72250162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302544);