Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,23,17)(7,18,21,33,22,32,36,34)(9,24,26,14,11,28,27,16)(13,29,30,25,19,35,31,20);(5,7,11,20,23,36,26,35,12,22,9,25,10,21,27,29)(6,13,16,18,15,30,24,33,8,19,14,32,17,31,28,34);(2,4)(5,8)(6,12)(7,19)(9,16)(10,15)(11,14)(13,22)(17,23)(18,25)(20,32)(21,30)(24,27)(26,28)(29,33)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321300, #72321310, #72321342, … |
| maximal normal subgroup of | #72321300, #72321310, #72321342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53652);