Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2023 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,16,18,33)(8,21)(9,14)(10,26)(11,15)(12,19,28,35)(13,25)(17,34,20,30)(22,32)(23,31,36,27)(24,29);(1,2,3,4)(7,17,18,20)(8,22)(12,27,28,31)(13,29)(16,30,33,34)(19,36,35,23)(21,32)(24,25);(5,7)(6,12)(8,23)(9,20)(10,17)(11,18)(13,30)(14,31)(15,28)(16,32)(19,24)(21,33)(22,36)(25,35)(26,27)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,d^c=di,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237285, #72238782, #72239707, … |
| maximal normal subgroup of | #72237285, #72238782, #72239707, … |
| maximal quotient of | #72237279 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55256);