Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1101 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)(6,11)(7,16)(8,19)(9,12)(13,26)(14,29)(18,21)(20,22)(23,28)(24,31)(27,30)(32,33);(5,6,12,25,10,11,9,15)(7,13,21,26,18,23,16,28)(8,14,19,31,20,24,22,29)(17,27,33,35,34,30,32,36);(2,4)(5,7,10,18)(6,13,11,23)(8,17,20,34)(9,21,12,16)(14,30,24,27)(15,26,25,28)(19,32,22,33)(29,36,31,35);(5,8)(6,14)(7,17)(9,22)(10,20)(11,24)(12,19)(13,27)(15,29)(16,32)(18,34)(21,33)(23,30)(25,31)(26,35)(28,36)Polycyclic presentation:
2^9|b^2=egh,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378637);