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 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,25)(7,13,18,29,19,30,26,28)(8,20,23,32,24,12,21,14)(17,33,34,27,35,31,22,36);(6,12,16,20)(7,17,19,35)(8,21)(9,10)(13,28)(14,15,32,25)(18,22,26,34)(23,24)(27,36)(29,30);(2,4);(5,7,11,19)(6,13,16,30)(8,22,24,34)(9,18,10,26)(12,27,20,36)(14,31,32,33)(15,29,25,28)(17,21,35,23)Polycyclic presentation:
2^9|a^2=g,b^2=gh,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 | #72250863, #72250867, #72250884, … |
| maximal normal subgroup of | #72250863, #72250867, #72250884, … |
| maximal quotient of | #72287706, #72309414, #72309601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405317);