Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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 | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17,20,34)(8,13,24,31)(9,22)(10,16)(11,29)(12,18,25,30)(14,27)(15,28)(19,23,21,35)(26,36)(32,33);(1,2)(3,4)(5,7,9,18,11,21,27,31,10,20,26,30,28,19,33,13)(6,12,14,17,15,8,22,23,16,25,32,34,29,24,36,35);(2,4)(5,8)(6,13)(7,14)(9,23)(10,24)(11,25)(12,28)(15,18)(16,31)(17,33)(19,36)(20,32)(21,22)(26,35)(27,34)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239850, #72239853, #72240855, … |
| maximal normal subgroup of | #72239850, #72239853, #72240855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55492);