Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136735 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1038 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)(7,17,21,35)(8,23)(9,13)(10,28)(11,16)(12,19,31,18)(14,26)(15,27)(20,32,24,34)(22,30,29,36)(25,33);(5,7)(6,12)(8,24)(9,18)(10,29)(11,21)(13,30)(14,17)(15,34)(16,31)(19,25)(20,26)(22,27)(23,35)(28,32)(33,36);(2,4)(7,18)(8,25)(9,26)(10,27)(12,17)(13,14)(15,28)(19,21)(20,22)(23,33)(24,29)(30,32)(31,35)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72226810, #72226813, … |
| maximal normal subgroup of | #72224903, #72226810, #72226813, … |
| maximal quotient of | #72224899, #72237264, #72265622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55119);