Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,12,15,31)(9,22)(10,26)(13,29,19,34)(14,30,33,35)(16,23,27,32)(17,24)(18,21)(20,36)(25,28);(5,6)(7,13)(8,19)(9,23)(10,27)(11,15)(12,28)(14,20)(16,26)(17,33)(18,35)(21,30)(22,31)(24,34)(25,32)(29,36);(1,2,3,4)(5,7,11,8)(6,13,15,19)(9,17,25,20)(10,18,26,21)(12,29,31,34)(14,23,33,32)(16,30,27,35)(22,24,28,36);(5,8,11,7)(6,14,15,33)(9,24,25,36)(10,18,26,21)(12,30,31,35)(13,23,19,32)(16,29,27,34)(17,28,20,22)Polycyclic presentation:
2^9|a^2=g,c^2=fi,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286631, #72286839, #72320360, … |
| maximal normal subgroup of | #72286631, #72286839, #72320360, … |
| maximal quotient of | #72287597, #72287750, #72309468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405063);