Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,17,21,19,22,27,15,16,20,26,25,28,24,18,23);(2,4)(5,6,8,12,9,11,7,10)(13,15,18,24,28,21,26,20,16,23,27,22,19,25,17,14);(6,11)(10,12)(14,20)(15,23)(21,25)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fg,c^b=cg,d^b=di,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34012);