Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | D4×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,11,13,15,21,17)(12,16,20,18,19,22,24,23);(1,2,4,6)(3,8,7,5)(10,15)(11,17)(16,22)(18,23);(9,11,14,15,13,17,21,10)(12,18,20,22,19,23,24,16);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24);(12,19)(16,22)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,c^b=ci,f^b=fh,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253111);