Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149785 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,13,10,15,22,32)(7,16,18,34,17,33,35,30)(9,14,24,31,25,12,21,29)(11,26,28,19,27,20,36,23);(1,2,3,4)(7,17)(11,27)(16,33)(18,35)(19,23)(20,26)(28,36)(30,34);(5,7,10,17)(6,11)(8,20,22,26)(9,23,25,19)(12,16)(13,30)(14,33)(15,27)(18,21,35,24)(28,29)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=h,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=eh,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59440);