Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1441 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)(7,17)(8,14)(9,12)(10,15)(11,16)(13,20)(18,26)(19,33)(21,30)(22,25)(23,29)(24,28)(27,35)(31,32)(34,36);(6,12)(7,18)(8,21)(9,10)(13,26)(14,29)(15,16)(17,28)(19,34)(22,23)(27,36)(32,35);(5,7,9,24,11,18,10,20)(6,13,15,26,16,28,12,17)(8,19,22,35,23,32,21,34)(14,27,30,33,29,36,25,31);(5,8)(6,14)(7,19)(9,22)(10,21)(11,23)(12,25)(13,27)(15,30)(16,29)(17,31)(18,32)(20,34)(24,35)(26,33)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293120);