Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483305 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1555 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #60646 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,11,22,28,7,13,23,30,4,10,19,27)(3,14,16,29,15,25,31,32,17,9,21,26,18,8,20,24);(1,3)(2,8)(4,15)(5,20)(6,18)(7,17)(9,11)(10,14)(12,26)(13,25)(16,19)(21,22)(23,31)(24,27)(28,32)(29,30);(1,3)(2,9)(4,18)(5,21)(6,15)(7,17)(8,11)(10,25)(12,29)(13,14)(16,23)(19,31)(20,22)(24,27)(26,30)(28,32);(1,4,7,6)(2,10,13,11)(3,15,17,18)(5,19,23,22)(8,14,25,9)(12,27,30,28)(16,31,21,20)(24,29,32,26)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,f^2=gh,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215848, #72215869, #72219679, … |
| maximal normal subgroup of | #72215848, #72215869, #72219679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390503);