Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(10,12)(14,17)(18,19)(21,22,25,30,26,28,24,29)(23,31,33,27,34,36,32,35);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,34)(22,27,28,35)(24,33,25,32)(29,36,30,31);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(22,28)(27,35)(29,30)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288469);