Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 | C2×#577 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3,5,6,8,7)(10,15)(11,13)(16,22)(17,21)(18,20)(23,24);(9,10,14,17)(11,15,13,21)(12,16,19,23)(18,22,20,24);(1,3,4,6,5,7,8,2)(9,11,14,13)(10,15,17,21)(12,18,19,20)(16,22,23,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24);(12,19)(16,23)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,f^b=fh,e^d=eh122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251770);