Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | C4×C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,12,15,13,11,14,16);(2,6)(3,4)(7,8)(17,18,19,21,20,22,23,24);(2,7)(6,8)(9,11,12,16,13,10,14,15)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cg,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72223570, C8×#1580, #72223574, … |
| maximal normal subgroup of | #72223570, C8×#1580, #72223574, … |
| maximal quotient of | #72223570, C2×#96892, #72223649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4088);