Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#41 | C22×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,14,19,15,18,32,37,16,20,33,21,13,29,31,39)(11,12,23,27,24,26,17,34,25,28,35,38,22,30,36,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,23,15,24,32,17,16,25,33,35,13,22,31,36)(10,12,19,27,18,26,37,34,20,28,21,38,29,30,39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,21)(13,30)(14,27)(15,26)(16,28)(18,35)(19,25)(20,36)(22,37)(23,29)(24,39)(31,40)(32,34)(33,38);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=e,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,f^c=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405582);