Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 | |
577 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)(9,10,14,19,13,20,23,36,15,21,29,33,32,18,38,40)(11,22,25,17,26,35,30,37,27,39,12,28,24,16,31,34);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,29)(13,26)(14,25)(15,27)(17,36)(18,39)(19,34)(20,22)(21,35)(23,30)(24,32)(28,40)(31,38)(33,37);(1,4,5,3)(2,6,7,8);(9,12)(10,17)(11,23)(13,31)(14,24)(15,25)(16,33)(18,34)(19,35)(20,37)(21,28)(22,40)(26,29)(27,38)(30,32)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,d^b=de,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405169);