Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33666 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(18,23)(19,21)(20,27)(24,32)(25,31)(26,29)(28,30);(17,18)(19,24)(20,28)(21,30)(22,25)(23,27)(26,31)(29,32);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19,22,21)(18,24,25,30)(20,26,29,27)(23,28,31,32);(2,7)(6,12)(8,14)(13,16)(17,19,22,21)(18,24,25,30)(20,26,29,27)(23,28,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72234320, #72234338, #72260313, … |
| maximal normal subgroup of | #72234320, #72234338, #72260313, … |
| maximal quotient of | #72320185, #72321045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391058);