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×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(13,16,17,24,18,25,15,22)(14,20,19,23,21,28,26,27);(6,9)(10,12)(16,25)(20,28)(22,24)(23,27);(5,7,8,11)(6,10,9,12)(16,25)(20,28)(22,24)(23,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,d^c=di,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72283181, #72284963 |
| maximal normal subgroup of | #72283181, #72284963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287597);