Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(11,17,18,19)(14,21,22,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,24)(20,23);(9,11,12,19,13,18,20,17)(10,14,16,24,15,22,23,21);(3,8)(5,7)(10,15)(14,22)(16,23)(21,24);(2,6)(5,7)(10,15)(14,22)(16,23)(21,24)Polycyclic presentation:
2^9|a^2=h,c^2=gh,g^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314467, #72315619, #72315677 |
| maximal normal subgroup of | #72314467, #72315619, #72315677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300565);