Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,22,26,23,27,20,24)(19,21,29,31,25,32,28,30);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,24,22,28,31,27,23,25,32,26,20,29,30,18);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,24,27,26)(19,28,25,29)(21,30,32,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch,d^b=di,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321857, #72321883, #72321909, … |
| maximal normal subgroup of | #72321857, #72321883, #72321909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33915);