Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4310 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)(3,4)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(2,4)(5,6)(7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,19)(18,23);(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(12,13)(16,17)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283148, #72317687, #72317709, … |
| maximal normal subgroup of | #72283148, #72317687, #72317709, … |
| maximal quotient of | #72283211, #72319875 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301091);