Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2676 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(10,11)(13,14,19,24,18,21,16,22)(15,17,23,28,25,20,26,27);(5,6,8,9)(7,11,12,10)(13,15,18,25)(14,20,21,17)(16,26,19,23)(22,28,24,27);(2,4)(14,21)(15,25)(22,24)(23,26);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,22,21,24)(15,23,25,26)(17,28,20,27);(13,17,19,28,18,20,16,27)(14,23,24,25,21,26,22,15)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291805);