Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| 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×#135 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,20,25,21,19,26,28,22,24,29,32,30,27,23,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(2,8)(7,15)(9,13)(10,16)(17,19,20,28,21,24,26,32,22,27,29,31,30,18,23,25);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(18,24)(19,27)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cgi,d^a=di,e^a=eg,e^b=eg,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72225551, #72227030, #72227031, … |
| maximal normal subgroup of | #72225551, #72227030, #72227031, … |
| maximal quotient of | #72225524, #72323193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390804);