Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
875 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)(7,15)(8,22)(9,13)(10,18)(11,16)(12,17)(14,25)(19,31)(20,29)(21,30)(23,28)(24,33)(26,32)(27,34)(35,36);(5,7,10,9,12,21,11,20)(6,13,16,15,17,29,18,30)(8,19,25,27,26,23,24,35)(14,31,22,36,33,28,32,34);(5,8,10,25,12,26,11,24)(6,14,16,22,17,33,18,32)(7,19,9,27,21,23,20,35)(13,28,15,34,29,31,30,36);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,29,17,30,18,13)(8,23,24,27,26,19,25,35)(14,31,32,34,33,28,22,36)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365322);