Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,16,10,17,27,15)(7,12,20,22,19,8,13,23)(9,14,26,31,24,30,25,32)(18,28,29,36,33,21,34,35);(2,4)(5,7,10,19)(6,12,17,8)(9,18,24,33)(11,13,27,20)(14,28,30,21)(15,22,16,23)(25,29,26,34)(31,35,32,36);(5,8)(6,13)(7,16)(9,21)(10,12)(11,23)(14,29)(15,19)(17,20)(18,32)(22,27)(24,28)(25,36)(26,35)(30,34)(31,33);(2,4)(5,9)(6,14)(7,18)(8,21)(10,24)(11,25)(12,28)(13,29)(15,31)(16,32)(17,30)(19,33)(20,34)(22,35)(23,36)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266741);