Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,19)(8,23,27,32)(9,13)(10,17)(11,16)(12,18)(14,26,33,25)(15,22)(20,28,35,36)(21,29)(24,31,34,30);(1,2,3,4)(5,7,10,9,12,22,11,21)(6,13,16,19,18,29,17,15)(8,24,25,20,27,34,26,35)(14,30,23,28,33,31,32,36);(5,8)(6,14)(7,20)(9,24)(10,26)(11,25)(12,27)(13,28)(15,31)(16,32)(17,23)(18,33)(19,30)(21,34)(22,35)(29,36);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,26,35,27,34,25,20)(14,31,32,28,33,30,23,36)Polycyclic presentation:
2^9|a^2=gh,b^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322665);