Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
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×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,14,20,15,21,30,18,16,22,29,38,13,19,12,27)(11,23,24,40,25,36,28,34,26,39,32,37,31,17,33,35);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,28)(13,31)(14,32)(15,25)(16,26)(18,37)(19,39)(20,34)(21,23)(22,36)(24,29)(27,40)(30,33)(35,38);(1,4,5,3)(2,6,7,8);(9,12,13,29,16,30,15,14)(10,18,19,20,22,27,21,38)(11,24,25,28,26,32,31,33)(17,34,23,37,36,35,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405231);