Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,13,17,14,16,12,18)(11,19,20,15,22,24,21,23);(5,6,7,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,24)(18,19);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239049 |
| maximal normal subgroup of | #72239049 |
| maximal quotient of | #72239058, #72239084 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261243);