Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,19,29,34,15,20,30,35,28,18,12,26)(11,21,23,16,25,32,39,33,22,37,38,31,24,40,27,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,27)(13,23)(14,25)(15,22)(17,32)(18,36)(19,33)(20,31)(21,26)(24,28)(29,39)(30,38)(34,37)(35,40);(11,22)(16,31)(21,37)(23,38)(24,25)(27,39)(32,40)(33,36)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31270);