Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | D4×#135 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,8,6)(9,10,14,20,15,22,31,21)(11,23,26,39,28,36,27,40)(12,29,32,18,13,33,35,19)(16,34,37,25,17,30,38,24);(5,8)(6,7)(9,11)(10,16)(12,30)(13,34)(14,26)(15,28)(17,22)(18,39)(19,40)(20,37)(21,38)(23,29)(24,32)(25,35)(27,31)(33,36);(1,3)(2,4)(5,6)(7,8)(9,11)(10,17)(12,30)(13,34)(14,26)(15,28)(16,22)(18,40)(19,39)(20,38)(21,37)(23,33)(24,32)(25,35)(27,31)(29,36)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=gi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=eg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39043);