Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(6,12)(7,16)(9,10)(14,26)(15,23)(17,31)(18,19)(20,30)(21,24)(25,28)(32,35)(33,34);(1,3,4,2)(5,6)(7,13)(8,20)(9,12)(10,23)(11,15)(14,21)(16,28)(17,32)(18,25)(19,29)(22,26)(24,30)(27,33)(31,36)(34,35);(5,7)(6,13)(8,17)(9,18)(10,16)(11,19)(12,25)(14,27)(15,29)(20,32)(21,33)(22,34)(23,28)(24,31)(26,35)(30,36);(5,8)(6,14)(7,17)(9,21)(10,24)(11,22)(12,26)(13,27)(15,30)(16,31)(18,33)(19,34)(20,23)(25,35)(28,32)(29,36);(5,9,11,10)(6,12,15,23)(7,18,19,16)(8,21,22,24)(13,25,29,28)(14,26,30,20)(17,33,34,31)(27,35,36,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,d^b=df,f^b=fi,g^c=gi,f^d=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288670);