Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,14,11,16,10,15)(7,17,19,12,21,28,20,29)(8,22,25,32,24,31,26,13)(18,27,23,30,34,36,35,33);(1,3,4,2)(5,7,10,20,11,21,9,19)(6,12,14,28,16,29,15,17)(8,23,26,18,24,35,25,34)(13,30,22,36,32,33,31,27);(2,3)(5,8,9,25,11,24,10,26)(6,13,15,31,16,32,14,22)(7,18,19,23,21,34,20,35)(12,27,17,33,29,36,28,30);(6,14,16,15)(8,24)(12,28,29,17)(13,31,32,22)(18,34)(23,35)(25,26)(27,33,36,30)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285236);