Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | |
1049 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)(7,18,21,28)(8,22)(9,13,20,29)(10,15)(11,16)(12,17)(14,24)(19,27,34,35)(23,36,33,30)(25,32)(26,31);(1,2)(3,4)(5,7)(6,13)(8,23)(9,10)(11,20)(12,21)(14,30)(15,28)(16,18)(17,29)(19,25)(22,35)(24,34)(26,33)(27,31)(32,36);(2,4)(5,8,10,24,12,26,11,25)(6,14,16,22,17,32,15,31)(7,19,20,33,21,34,9,23)(13,27,28,36,29,35,18,30);(5,9,11,21,12,20,10,7)(6,13,16,28,17,29,15,18)(8,23,25,34,26,33,24,19)(14,27,22,36,32,35,31,30)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291527);