Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,22)(9,13)(10,18)(11,14)(12,16)(15,20)(19,29)(21,27)(23,32)(24,35)(25,36)(26,30)(28,33)(31,34);(1,2,4,3)(6,13)(7,19)(11,21)(15,28)(16,18)(20,33)(26,34)(30,31);(1,3,4,2)(5,7,10,11,9,19,12,21)(6,14,16,17,13,27,18,29)(8,20,24,26,25,33,23,34)(15,22,31,32,28,36,30,35);(5,8)(6,15)(7,20)(9,25)(10,24)(11,26)(12,23)(13,28)(14,22)(16,31)(17,32)(18,30)(19,33)(21,34)(27,36)(29,35)Polycyclic presentation:
2^9|b^2=fh,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419387);