Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11)(7,15)(8,20)(9,23)(12,26)(13,29)(16,28)(17,33)(18,19)(21,22)(24,31)(35,36);(1,2)(3,4)(5,6,9,14,10,11,23,25)(7,16,15,26,19,12,18,28)(8,13,21,29,22,24,20,31)(17,30,32,35,33,27,34,36);(2,4)(5,7)(6,12,11,16)(8,17)(9,18)(10,19)(13,30,24,27)(14,26,25,28)(15,23)(20,32)(21,34)(22,33)(29,36,31,35);(5,8,10,22)(6,13,11,24)(7,17,19,33)(9,21,23,20)(12,27,16,30)(14,29,25,31)(15,32,18,34)(26,35,28,36)Polycyclic presentation:
2^9|b^2=f,c^2=e,d^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,g^c=gi,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, 363019);