Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,17)(8,21,22,30)(10,16,11,15)(13,20)(14,23,26,24)(18,29)(19,33)(25,27)(28,32)(31,34)(35,36);(1,2)(3,4)(6,12)(7,18,20,25)(8,22)(13,27,17,29)(15,16)(19,32,35,34)(23,24)(28,36,31,33);(2,4)(5,7)(6,13)(8,19)(9,20)(10,25)(11,18)(12,17)(14,28)(15,27)(16,29)(21,36)(22,35)(23,32)(24,34)(26,31)(30,33);(5,8,9,22)(6,14,12,26)(7,19,20,35)(10,24,11,23)(13,28,17,31)(15,21,16,30)(18,32,25,34)(27,36,29,33)Polycyclic presentation:
2^9|a^2=e,b^2=ei,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325102);