Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(3,7)(5,8)(9,10)(11,17,19,24)(12,15)(13,16)(14,18,22,21)(20,23);(2,5,6,8)(12,20)(14,22)(15,23)(18,21);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,21)(13,19)(15,24)(16,22)(17,23)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233844, #72233850, #72233851, … |
| maximal normal subgroup of | #72233844, #72233850, #72233851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23567);