Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| 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 | #33519 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
752 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)(6,11,14,28)(7,16)(9,18)(12,23,15,26)(13,27,30,33)(17,22)(19,29,35,31)(20,36)(24,34)(25,32);(5,6,7,12,10,14,16,15)(8,19,22,27,21,35,17,33)(9,23,18,11,25,26,32,28)(13,24,31,20,30,36,29,34);(2,4)(5,7,10,16)(6,12,14,15)(8,17,21,22)(9,18,25,32)(11,26,28,23)(13,29,30,31)(19,33,35,27)(20,24,34,36);(5,8,10,21)(6,13,14,30)(7,17,16,22)(9,24,25,36)(11,27,28,33)(12,29,15,31)(18,34,32,20)(19,23,35,26)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,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=eh,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286595, #72286631, #72286724, … |
| maximal normal subgroup of | #72286595, #72286631, #72286724, … |
| maximal quotient of | #72287279, #72287378, #72287493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405315);