Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D8×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
522 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,23,38)(12,24,27,37)(13,28,26,16)(15,30,32,21)(17,34,36,25)(20,33,39,31)(22,40,29,35);(9,11,14,23)(10,15,18,32)(12,20,27,39)(13,29,26,22)(16,33,28,31)(17,35,36,40)(19,37,38,24)(21,34,30,25);(2,5)(3,8)(6,7);(10,16)(11,20)(12,25)(13,26)(15,19)(17,36)(18,28)(21,22)(23,39)(27,34)(29,30)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fh,g^a=gi,d^b=de,e^b=eg,g^b=gi,f^c=fh,e^d=eg,g^d=gi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72309422 |
| maximal normal subgroup of | #72309422 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399718);