Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,17,11,33)(7,18,21,26,22,35,23,30)(8,24,27,34,28,20,13,14)(9,29,31,25,32,36,19,15);(5,7,19,34,11,23,32,24,12,22,31,14,10,21,9,20)(6,13,29,26,16,8,25,35,17,27,36,30,33,28,15,18);(2,4)(5,8)(6,14)(7,15)(9,30)(10,13)(11,27)(12,28)(16,20)(17,34)(18,19)(21,36)(22,25)(23,29)(24,33)(26,32)(31,35)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250063, #72250075, #72250104, … |
| maximal normal subgroup of | #72250063, #72250075, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55550);