Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4253 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)(3,5)(4,6)(7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(2,5)(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(2,6)(5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(12,21)(15,20)(18,24)(22,23);(10,16)(14,17)(15,20)(22,23)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281866, #72281894, #72281920, … |
| maximal normal subgroup of | #72281866, #72281894, #72281920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250851);