Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| 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 | |
1314 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)(8,13)(9,11,18,14)(12,20)(17,26)(19,33,34,31)(21,30)(22,29)(23,28)(24,25)(27,35,36,32);(6,11)(7,17)(8,22)(9,18)(12,26)(13,28)(14,15)(19,34)(20,21)(23,24)(27,31)(33,36);(2,4)(5,7,18,20,10,21,9,17)(6,12,11,16,15,26,14,30)(8,19,22,35,23,34,24,32)(13,27,25,31,28,36,29,33);(2,4)(5,8)(6,13)(7,19)(9,24)(10,23)(11,25)(12,27)(14,29)(15,28)(16,31)(17,32)(18,22)(20,35)(21,34)(26,36)(30,33)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,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, 295584);