Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,18)(8,13)(9,10)(14,29)(15,26)(16,25)(17,27)(19,33)(20,23)(21,22)(24,30)(31,36)(32,35);(1,2)(3,4)(5,6,13,26,10,16,22,27,11,17,21,25,9,15,8,12)(7,14,28,29,18,30,33,35,20,31,34,36,23,32,19,24);(2,4)(5,7)(6,14)(8,19)(9,23)(10,18)(11,20)(12,24)(13,28)(15,32)(16,30)(17,31)(21,34)(22,33)(25,36)(26,29)(27,35);(5,6,13,26,10,16,22,27,11,17,21,25,9,15,8,12)(7,14,28,29,18,30,33,35,20,31,34,36,23,32,19,24)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239838, #72321154, #72321425, … |
| maximal normal subgroup of | #72239838, #72321154, #72321425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408039);