Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,12,17,13,16,21,24)(11,14,19,22,18,15,20,23);(5,6)(7,8)(9,11)(10,14,16,15)(12,20)(13,18)(17,23,24,22)(19,21);(2,4)(6,8)(9,11)(10,15)(12,20)(13,18)(14,16)(17,22)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215689, #72215834, #72216103, … |
| maximal normal subgroup of | #72215689, #72215834, #72216103, … |
| maximal quotient of | #72215673, #72215810, #72226736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30788);