Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1094 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)(5,6,9,15,10,12,25,26,11,16,23,33,8,14,22,31)(7,17,20,34,19,30,35,32,21,28,24,36,18,13,27,29);(2,4)(6,12,16,14)(9,23)(13,17,30,28)(15,31,33,26)(20,24)(22,25)(27,35)(29,36,32,34);(5,7)(6,13)(8,18)(9,24)(10,19)(11,21)(12,17)(14,28)(15,32)(16,30)(20,23)(22,35)(25,27)(26,36)(29,33)(31,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,f^b=fi,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72240007, #72240008, #72240009, … |
| maximal normal subgroup of | #72240007, #72240008, #72240009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53581);