Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18,23,26)(19,21,25,22)(20,28,32,27)(24,30,29,31);(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,32)(27,30)(28,31);(2,4)(9,11)(10,14)(12,15)(13,16)(17,20,23,32)(18,24,26,29)(19,27,25,28)(21,30,22,31);(6,8)(11,15)(14,16)(17,21,23,22)(18,25,26,19)(20,30,32,31)(24,28,29,27)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361406);