Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,13,16,14,17,12,18)(11,19,20,23,21,15,22,24);(1,2)(3,4)(5,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(2,4)(10,16)(12,13)(15,23)(17,18)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72251167, #72251172, #72251201, … |
| maximal normal subgroup of | #72251167, #72251172, #72251201, … |
| maximal quotient of | C2×#123547, #72338231, #72338242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30743);