Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129435 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1189 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(5,6)(7,15)(8,19)(9,23,24,13)(10,14)(11,17,26,35)(12,21,29,20)(16,25)(18,31)(22,28)(27,34)(30,33,36,32);(1,2)(3,4)(5,7,10,18)(6,11,14,26)(8,16,22,34)(9,17,24,35)(12,27,29,25)(13,15,23,31)(19,36,28,30)(20,32,21,33);(2,4)(5,8)(6,12)(7,16)(9,21)(10,22)(11,25)(13,19)(14,29)(15,30)(17,33)(18,34)(20,24)(23,28)(26,27)(31,36)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,g^b=gh,d^c=di,g^c=gh,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303133, #72303135, #72303152, … |
| maximal normal subgroup of | #72303133, #72303135, #72303152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54929);