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×#553 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2189 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)(6,9)(10,12)(13,14)(15,23)(16,21)(17,22)(18,20)(19,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,27)(22,23);(5,7,8,11)(6,10,9,12)(14,20)(15,24)(16,18)(19,27)(21,22)(23,28);(6,9)(10,12)(15,24)(19,23)(25,26)(27,28);(1,3,4,2)(13,16,17,18)(14,21,22,20)(15,25,24,26)(19,27,23,28)Polycyclic presentation:
2^9|a^2=h,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291586);