Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,24)(21,33)(22,34)(23,30)(25,31)(26,32)(27,38)(29,36)(35,39)(37,40);(9,11,14,24,15,22,13,23)(10,16,18,20,19,30,28,34)(12,21,27,36,25,37,26,35)(17,29,32,33,31,39,38,40);(2,4)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,37)(23,35)(24,36)(28,38)(30,39)(34,40);(5,7,8,6)(11,22)(12,25)(16,30)(17,31)(20,34)(23,24)(26,27)(32,38);(9,13,15,14)(10,18,19,28)(11,23,22,24)(12,26,25,27)(16,20,30,34)(17,32,31,38)(21,35,37,36)(29,33,39,40)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72343393, #72343397, #72343405, … |
| maximal normal subgroup of | #72343393, #72343397, #72343405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280065);