Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,23,24,20)(19,22);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,23)(19,24);(1,3,4,8,5,2,6,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17517);