Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12,19,8)(9,14,23,31)(10,16)(11,15)(13,22,20,26)(17,25)(18,28)(21,33)(24,30,27,32)(29,36)(34,35);(2,4)(5,7,11,13,10,19,25,20)(6,12,15,22,16,8,17,26)(9,18,27,29,23,33,24,34)(14,28,32,36,31,21,30,35);(1,2,3,4)(5,8)(6,13)(7,17)(9,21)(10,12)(11,26)(14,29)(15,19)(16,20)(18,30)(22,25)(23,28)(24,36)(27,35)(31,34)(32,33);(5,9)(6,14)(7,18)(8,21)(10,23)(11,27)(12,28)(13,29)(15,32)(16,31)(17,30)(19,33)(20,34)(22,36)(24,25)(26,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277304);