Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 36 points:
(2,4)(5,6)(7,15,18,30,19,33,16,32)(8,11)(9,22)(10,14)(12,17,28,24,25,36,27,34)(13,20)(21,26)(23,29)(31,35);(1,2,3,4)(6,11)(9,23)(10,21)(12,25)(13,29)(15,30)(16,18)(17,34)(20,35)(22,31)(24,36)(32,33);(5,7)(6,12)(8,19)(9,24)(10,18)(11,25)(13,30)(14,28)(15,29)(16,21)(17,35)(20,34)(22,32)(23,36)(26,27)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224798, #72226756, #72226757, … |
| maximal normal subgroup of | #72224798, #72226756, #72226757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44065);