Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1126 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:
(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,17,18,26,21,34,20,33)(8,13,22,29,23,30,24,32)(12,25,27,36,28,35,31,19);(2,4)(5,7)(6,12)(8,19)(9,18)(10,20)(11,21)(13,17)(14,31)(15,27)(16,28)(22,35)(23,36)(24,25)(26,29)(30,34)(32,33);(6,13)(7,18)(9,10)(12,26)(14,32)(15,29)(16,30)(17,31)(19,35)(20,21)(22,24)(25,36)(27,34)(28,33)Polycyclic presentation:
2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gh,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72337423, #72337424, #72337427, … |
| maximal normal subgroup of | #72337423, #72337424, #72337427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54921);