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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 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)(5,6,11,17,12,18,8,14)(7,19,20,32,24,26,23,36)(9,15,16,10,27,29,25,30)(13,28,31,22,21,34,33,35);(5,7)(6,13,18,21)(8,20)(9,26,27,19)(10,28)(11,23)(12,24)(14,31,17,33)(15,22)(16,32,25,36)(29,35)(30,34);(2,4)(5,8,12,11)(6,14,18,17)(7,20,24,23)(9,25,27,16)(10,15,30,29)(13,31,21,33)(19,32,26,36)(22,34,35,28);(5,9,12,27)(6,15,18,29)(7,21,24,13)(8,25,11,16)(10,14,30,17)(19,34,26,28)(20,33,23,31)(22,32,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=h,g^2=i,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 | #72286580, #72286631, #72286696, … |
| maximal normal subgroup of | #72286580, #72286631, #72286696, … |
| maximal quotient of | #72287131, #72287197, #72287422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405319);