Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
898 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)(5,6,11,15,10,16,9,13)(7,17,20,12,21,28,18,26)(8,14,24,29,25,32,23,31)(19,33,22,36,35,30,34,27);(5,7)(6,12)(8,22)(9,18)(10,21)(11,20)(13,17)(14,30)(15,28)(16,26)(19,23)(24,35)(25,34)(27,29)(31,36)(32,33);(6,13,16,15)(7,18,21,20)(12,26)(14,29,32,31)(17,28)(19,22,35,34)(27,36)(30,33);(2,4)(5,8,9,24,10,25,11,23)(6,14,15,31,16,32,13,29)(7,19,18,22,21,35,20,34)(12,27,28,30,26,36,17,33)Polycyclic presentation:
2^9|a^2=efg,c^2=fgi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353060);