Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #35190 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,13)(12,20,22,16)(14,17)(15,18)(19,21,24,23);(2,6,8,7)(9,11,13,10,15,17,14,18)(12,19,21,16,22,24,23,20);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,24)(18,20);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,24,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^b=fi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108822);