Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9383 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,20,21,24)(12,16,22,23)(13,17)(14,18)(15,19);(2,6,8,7)(9,11,14,23,15,21,13,16)(10,12,18,24,19,22,17,20);(9,12)(10,16)(11,18)(13,20)(14,24)(15,22)(17,21)(19,23);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,17,19,18)(11,16,21,23)(12,20,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107637);