Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,12,15,14,18,13,17)(11,19,21,23,20,24,22,16);(1,2,3,7,4,6,8,5)(10,15,18,17)(16,19,23,24);(9,11,13,22,14,20,12,21)(10,16,15,19,18,23,17,24);(11,20)(16,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112316);