Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C2×#342 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 242 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:
(5,6,9,14,11,16,10,15)(7,17,20,33,18,32,21,34)(8,22,23,31,24,13,25,29)(12,26,28,19,27,36,30,35);(5,7,10,21,11,18,9,20)(6,12,15,28,16,27,14,30)(8,19,23,36,24,35,25,26)(13,17,29,34,22,32,31,33);(1,2,3,4)(7,18)(12,27)(17,32)(19,35)(20,21)(26,36)(28,30)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bd,d^a=dh,h^a=hi,c^b=ci,e^b=eh,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312327, #72312339, #72312563, … |
| maximal normal subgroup of | #72312327, #72312339, #72312563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53547);