Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115240 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,17,13,15,23,16)(11,18,21,24,22,14,20,19);(5,8)(9,11)(10,14)(12,20)(13,22)(15,18)(16,19)(17,24)(21,23);(1,3)(2,5)(4,7)(6,8)(9,10)(11,19)(12,17)(13,15)(14,20)(16,23)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72267261, #72267277, #72267278 |
| maximal normal subgroup of | #72267261, #72267277, #72267278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22440);