Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(6,11,14,21)(12,23,25,31)(13,18,26,32)(24,27,34,35);(5,6,9,21,10,14,22,11)(7,12,17,31,16,25,30,23)(8,18,20,13,19,32,33,26)(15,27,28,24,29,35,36,34);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,23)(13,24)(14,25)(18,27)(19,29)(20,28)(21,31)(22,30)(26,34)(32,35)(33,36);(5,8,9,20,10,19,22,33)(6,13,11,18,14,26,21,32)(7,15,17,28,16,29,30,36)(12,24,23,27,25,34,31,35);(7,16)(8,19)(12,25)(13,26)(17,30)(18,32)(20,33)(23,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289115);