Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220190 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,16,11,18)(7,19,22,32,23,13,21,33)(8,24,26,34,28,35,27,14)(9,29,30,31,20,15,25,36);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,17,32,16,19,18,33)(8,25,26,20,28,30,27,9)(14,15,24,31,34,29,35,36);(2,4)(5,8,12,28)(6,14,16,34)(7,20,23,9)(10,27,11,26)(13,31,19,36)(15,33,29,32)(17,35,18,24)(21,25,22,30);(5,9)(6,15)(7,8)(10,25)(11,30)(12,20)(13,24)(14,33)(16,29)(17,31)(18,36)(19,35)(21,26)(22,27)(23,28)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267483);