Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#397 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1174 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)(5,6,10,14)(7,15,18,28)(8,19,21,12)(9,22,24,13)(11,23,26,17)(16,30,32,29)(20,27,36,34)(25,31,33,35);(5,7)(6,11)(8,16,21,32)(9,23)(10,18)(12,25,19,33)(13,28)(14,26)(15,22)(17,24)(20,35,36,31)(27,30,34,29);(5,8,9,20,10,21,24,36)(6,12,13,27,14,19,22,34)(7,16,17,31,18,32,23,35)(11,25,15,29,26,33,28,30);(2,4)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303810);