Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,19,16,20,15,24)(11,21,18,13,22,23,12,17);(1,2,3,4)(9,11,13,10,15,18,21,19,16,22,17,20,14,12,23,24);(6,8)(9,12)(10,17)(11,14)(13,20)(15,22)(16,18)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319720, #72319735, #72319757, … |
| maximal normal subgroup of | #72319720, #72319735, #72319757, … |
| maximal quotient of | #72322462 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55571);