Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×AGL(1,ℤ/8) | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ #2162 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,20,24,21,19,26,27,22,25,29,32,30,28,23,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19,20,27,21,25,26,32,22,28,29,31,30,18,23,24);(3,10)(7,14)(11,16)(13,15)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,e^b=eg,d^c=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320765, #72322030, #72322081 |
| maximal normal subgroup of | #72320765, #72322030, #72322081 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390863);