Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| 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 | |
1403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,16,11,13,9,15)(7,12,19,25,20,27,17,29)(8,21,22,30,23,14,24,28)(18,32,31,35,34,26,33,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,17)(10,19)(11,20)(13,27)(14,26)(15,29)(16,25)(21,32)(22,31)(23,34)(24,33)(28,36)(30,35);(6,13)(7,17)(9,10)(12,25)(14,21)(18,31)(19,20)(22,24)(26,35)(27,29)(32,36)(33,34);(1,3,4,2)(5,8,11,23)(6,14,13,21)(7,18,20,34)(9,22,10,24)(12,26,27,32)(15,28,16,30)(17,31,19,33)(25,35,29,36)Polycyclic presentation:
2^9|a^2=h,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290683);