Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1050 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)(5,6,9,16,10,14,23,30,11,15,24,31,8,20,22,33)(7,13,18,29,17,12,25,27,19,28,32,36,21,26,34,35);(5,7)(6,12)(8,21)(9,18)(10,17)(11,19)(13,20)(14,28)(15,26)(16,27)(22,34)(23,25)(24,32)(29,33)(30,36)(31,35);(2,4)(5,7)(6,13)(8,21)(9,18)(10,17)(11,19)(12,14)(15,28)(16,29)(20,26)(22,34)(23,25)(24,32)(27,30)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281194, #72281222, #72281244, … |
| maximal normal subgroup of | #72281194, #72281222, #72281244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54739);