Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,7,12,15)(8,13,23,29,22,17,21,19)(9,24,27,18,26,30,25,14)(20,32,33,28,34,36,35,31);(1,2,3,4)(5,7)(6,11)(8,19)(9,18)(10,15)(12,16)(13,23)(14,26)(17,21)(20,32)(22,29)(24,25)(27,30)(28,33)(31,35)(34,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,28)(15,29)(16,19)(18,31)(24,32)(25,33)(26,34)(27,35)(30,36);(5,9,11,26)(6,14,7,18)(8,20,22,34)(10,27,12,25)(13,28,17,31)(15,30,16,24)(19,32,29,36)(21,35,23,33)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273929);