Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 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)(6,12)(7,16)(8,21)(9,10)(13,26)(14,29)(17,28)(18,33)(19,20)(22,23)(25,30)(27,32);(5,6,9,15,11,12,10,24)(7,17,16,26,20,13,19,28)(8,14,23,30,22,25,21,29)(18,32,34,36,33,27,31,35);(5,7,11,20)(6,13,12,17)(8,18,22,33)(9,19,10,16)(14,27,25,32)(15,26,24,28)(21,34,23,31)(29,35,30,36);(2,4)(5,8)(6,14,12,25)(7,18)(9,21)(10,23)(11,22)(13,27,17,32)(15,29,24,30)(16,31)(19,34)(20,33)(26,35,28,36)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355843);