Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(5,6,7,10,8,11,12,9);(6,9)(7,12)(10,11)(13,14,15,19,16,20,21,26,17,18,22,24,23,25,27,28);(1,3,4,2)(14,18)(19,24)(20,25)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^b=ci,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72216252, #72216276, #72216306, … |
| maximal normal subgroup of | #72216252, #72216276, #72216306, … |
| maximal quotient of | #72216243, #72225375, #72225412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36289);