Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,10)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,3,4,2)(5,6,9,10)(7,11,8,12);(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(5,8,9,7)(6,11,10,12);(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289390);