Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,28,33)(7,16,19,31,20,32,27,12)(8,13,22,9,23,25,36,26)(17,24,30,29,21,34,35,18);(5,7)(6,12)(8,21)(9,24)(10,27)(11,20)(13,29)(14,32)(15,31)(16,33)(17,23)(18,25)(19,28)(22,30)(26,34)(35,36);(5,8)(6,13)(7,17,20,21)(9,14)(10,22)(11,23)(12,29,31,18)(15,25)(16,34,32,24)(19,30,27,35)(26,33)(28,36);(2,4)(5,9,10,25,11,26,28,13)(6,8,14,22,15,23,33,36)(7,18,19,24,20,29,27,34)(12,30,16,21,31,35,32,17)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323003 |
| maximal normal subgroup of | #72323003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331031);