Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
530 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)(7,15,17,11)(8,13)(9,14)(10,12)(16,25,19,23)(18,29,32,24)(20,26)(21,28)(22,27)(30,34)(31,36,33,35);(5,7,8,18,9,19,21,33,10,17,20,32,22,16,30,31)(6,11,13,24,14,25,28,36,12,15,26,29,27,23,34,35);(2,4)(7,16,17,19)(8,20)(11,23,15,25)(13,26)(18,33,32,31)(21,30)(24,36,29,35)(28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,f^b=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239244, #72239251, #72239284, … |
| maximal normal subgroup of | #72239244, #72239251, #72239284, … |
| maximal quotient of | #72239342, C2×#147126 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54322);