Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,16,14,13,11,18)(12,17,22,23,21,20,19,24);(1,2,3,4)(9,11,14,15)(10,16,13,18)(12,19,21,22)(17,23,20,24);(6,8)(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24);(9,13,15,18,14,10,11,16)(12,20,22,24,21,17,19,23);(12,21)(17,20)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^b=di,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216198 |
| maximal normal subgroup of | #72216198 |
| maximal quotient of | #72215932 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280120);