Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,19)(12,20)(13,17)(14,23)(15,18)(16,22)(21,24);(1,2)(3,4)(9,11,15,21,13,20,14,16)(10,12,18,22,17,19,23,24);(5,6,8,7)(9,12)(10,16)(11,18)(13,19)(14,24)(15,22)(17,21)(20,23);(2,4)(5,7,8,6)(10,17)(11,20)(16,21)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,f^c=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263809);