Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395775 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1798 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)(5,6,10,15,11,16,9,14)(7,17,20,12,21,28,19,29)(8,22,24,30,25,31,26,13)(18,33,34,36,23,27,35,32);(5,7,11,21)(6,12,16,29)(8,23,25,18)(9,20,10,19)(13,27,30,33)(14,28,15,17)(22,36,31,32)(24,34,26,35);(5,8,9,24,11,25,10,26)(6,13,14,22,16,30,15,31)(7,18,19,34,21,23,20,35)(12,27,17,32,29,33,28,36);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 302972);