Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| 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 | |
885 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)(7,19)(8,23)(9,13)(10,17)(11,16)(12,18)(14,25)(15,22)(20,29)(21,30)(24,31)(26,34)(27,33)(28,32)(35,36);(5,7,11,21,12,22,10,9)(6,13,17,15,18,30,16,19)(8,20,25,36,26,24,28,27)(14,31,23,35,32,29,34,33);(5,8,11,25,12,26,10,28)(6,14,17,23,18,32,16,34)(7,20,21,36,22,24,9,27)(13,29,15,33,30,31,19,35);(2,4)(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,30)(8,24,25,27,26,20,28,36)(14,29,23,33,32,31,34,35)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^c=de,e^c=ei,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365321);