Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2192 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,18)(8,12)(9,23)(10,17)(11,16)(13,20)(14,21)(15,26)(19,33)(22,32)(24,34)(25,30)(27,29)(28,35)(31,36);(1,2,3,4)(6,12)(13,28)(14,29)(15,30)(16,17)(18,33)(23,32)(31,34);(5,7)(6,13)(8,19)(9,24)(10,21)(11,27)(12,28)(14,30)(15,29)(16,31)(17,34)(18,23)(20,26)(22,36)(25,35)(32,33);(5,7)(6,14)(8,19)(9,20)(10,27)(11,21)(12,29)(13,30)(15,28)(16,33)(17,18)(22,35)(23,34)(24,26)(25,36)(31,32)Polycyclic presentation:
2^9|b^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419099);