Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136643 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,15,11,16,10,25)(7,12,17,28,20,29,19,27)(8,21,22,31,24,13,23,14)(18,33,32,36,34,30,35,26);(1,2,4,3)(5,7)(6,12)(8,18)(9,17)(10,19)(11,20)(13,30)(14,26)(15,28)(16,29)(21,33)(22,32)(23,35)(24,34)(25,27)(31,36);(1,3,4,2)(6,13)(7,17)(8,22)(9,10)(12,26)(14,25)(15,31)(16,21)(18,34)(19,20)(23,24)(27,33)(28,30)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321689, #72321693, #72321720, … |
| maximal normal subgroup of | #72321689, #72321693, #72321720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55914);