Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
986 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)(3,7)(4,6)(5,8)(9,10,13,17,14,12,16,21)(11,18,20,24,19,15,22,23);(1,3)(2,5,6,7)(4,8)(9,11)(10,15)(12,18)(13,22)(14,19)(16,20)(17,24)(21,23);(9,12,13,21,14,10,16,17)(11,18,22,23,19,15,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224724, #72224728, #72224729, … |
| maximal normal subgroup of | #72224724, #72224728, #72224729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17607);