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) = C28 |
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×D4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1919 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)(15,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28);(5,6,9,10)(7,12,8,11)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,25)(24,28);(14,19)(15,22)(16,24)(18,25)(20,21)(26,27);(5,7,9,8)(6,11,10,12);(1,3,4,2)(5,8,9,7)(6,11,10,12)(14,20)(18,26)(19,21)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,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, 6292299);