Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,14,10,25)(7,16,19,33,18,32,26,30)(8,20,22,31,24,35,23,13)(12,21,28,34,27,36,29,17);(5,7)(6,12)(8,21)(9,19)(10,26)(11,18)(13,30)(14,27)(15,28)(16,20)(17,23)(22,34)(24,36)(25,29)(31,33)(32,35);(2,4)(12,27)(16,32)(28,29)(30,33)Polycyclic presentation:
2^9|a^2=g,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231484, #72231505, #72231508, … |
| maximal normal subgroup of | #72231484, #72231505, #72231508, … |
| maximal quotient of | #72265634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54839);