Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,17)(18,20)(21,22,23,27,25,28,24,26);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(22,26)(23,24)(27,28);(2,4)(14,17)(18,19)(22,27)(23,24)(26,28);(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^9|a^2=e,b^2=efh,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323574);