Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,19)(8,14,25,31)(9,15)(10,17)(11,18)(12,16)(13,21)(20,33,23,29)(22,28)(24,32,26,30)(27,35,36,34);(1,2)(3,4)(5,7,12,22,11,9,10,21)(6,13,17,15,18,28,16,19)(8,20,26,35,25,23,24,34)(14,27,32,29,31,36,30,33);(5,8)(6,14)(7,20)(9,23)(10,24)(11,25)(12,26)(13,27)(15,29)(16,30)(17,32)(18,31)(19,33)(21,34)(22,35)(28,36);(2,4)(5,9,11,7)(6,15,18,19)(8,23,25,20)(10,21,12,22)(13,16,28,17)(14,29,31,33)(24,34,26,35)(27,30,36,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=de,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263301);