Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480670 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,15)(12,19,20,14)(13,16)(17,22)(18,24,23,21);(2,6)(3,4)(10,14)(15,21)(16,19)(22,24);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(2,6)(7,8)(11,17)(15,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fg,d^c=di,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72234733, #72234734, #72234740, … |
| maximal normal subgroup of | #72234733, #72234734, #72234740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387917);