Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,17,11,15,28,30,12,16,29,36,9,27,19,34)(7,18,8,23,21,32,25,14,22,31,26,35,20,13,24,33);(1,2,3,4)(5,7,9,20,12,22,11,21)(6,13,15,18,16,32,27,31)(8,19,24,29,26,28,25,10)(14,34,35,17,33,30,23,36);(5,8)(6,14)(7,19)(9,24)(10,21)(11,25)(12,26)(13,30)(15,35)(16,33)(17,31)(18,36)(20,29)(22,28)(23,27)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320946, #72321051, #72321100, … |
| maximal normal subgroup of | #72320946, #72321051, #72321100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55178);