Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
801 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)(5,6,10,15)(7,16)(8,21)(9,11,17,14)(12,20)(13,22)(18,32,34,36)(19,26)(23,30)(24,27)(25,29)(28,35,31,33);(6,11,15,14)(8,22,23,25)(12,27,16,26)(13,29)(18,33,34,35)(21,30)(28,31)(32,36);(5,7,17,19,10,20,9,24)(6,12,11,26,15,16,14,27)(8,18,22,35,23,34,25,33)(13,28,21,36,29,31,30,32);(5,8)(6,13)(7,18)(9,25)(10,23)(11,21)(12,28)(14,30)(15,29)(16,31)(17,22)(19,35)(20,34)(24,33)(26,36)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=efg,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294417);