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 | C2×#3347 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2017 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)(13,14,17,20)(15,18,23,26)(16,19,24,21)(22,25,28,27);(5,6)(7,11)(8,12)(9,10)(13,15,17,23)(14,18,20,26)(16,22,24,28)(19,25,21,27);(5,7,9,8)(6,11,10,12)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27);(5,8,9,7)(6,11,10,12)(14,20)(18,26)(19,21)(25,27);(1,3,4,2)(14,20)(18,26)(19,21)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6291998);