Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16)(7,17,21,27)(8,22,25,29)(9,15,10,14)(12,20,28,18)(13,23,30,24)(19,26,34,36)(31,32,35,33);(5,7,10,20,11,21,9,18)(6,12,15,27,16,28,14,17)(8,19,23,33,25,34,24,32)(13,26,29,35,30,36,22,31);(1,2,4,3)(7,18)(8,23)(9,10)(12,17)(13,29)(14,15)(20,21)(22,30)(24,25)(27,28)(31,35)(32,33);(1,3,4,2)(5,8)(6,13)(7,19)(9,24)(10,23)(11,25)(12,26)(14,22)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(27,35)(28,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373837);