Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17)(8,21)(9,12,10,15)(13,25)(14,24)(18,29)(19,31,34,36)(20,28)(22,26)(23,30)(27,32,35,33);(2,4)(6,12)(7,18)(8,22)(9,10)(14,21)(15,16)(17,28)(20,25)(23,24)(27,31)(32,33)(35,36);(1,2,3,4)(5,7)(6,13)(8,19)(9,18)(10,25)(11,20)(12,17)(14,27)(15,28)(16,29)(21,35)(22,33)(23,32)(24,34)(26,31)(30,36);(5,8)(6,14)(7,19)(9,23)(10,22)(11,24)(12,26)(13,27)(15,30)(16,21)(17,31)(18,32)(20,34)(25,33)(28,36)(29,35)Polycyclic presentation:
2^9|a^2=h,c^2=eghi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297621);