Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,28,30,25)(20,26,21,24)(23,31,32,29);(1,2)(3,4)(17,19,21,29,22,30,20,31)(18,23,26,28,27,32,24,25);(2,4)(10,13)(14,16)(17,20,22,21)(18,24,27,26)(19,29,30,31)(23,28,32,25);(6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,25)(20,29)(21,31)(22,30)(23,24)(26,32)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335608);