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 | |
528 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(10,15)(11,20)(12,21)(13,22)(16,27)(17,23)(18,33)(19,31)(25,26)(28,38)(29,30)(32,37)(34,40)(36,39);(9,10,14,18)(11,16,20,19)(12,23,25,39)(13,27,29,31)(15,30,33,22)(17,35,36,24)(21,32,26,40)(28,37,38,34);(2,5)(3,8)(6,7)(9,11,14,20)(10,16,18,19)(12,21,25,26)(13,22,29,30)(15,31,33,27)(17,34,36,37)(23,32,39,40)(24,38,35,28);(9,12,14,25)(10,17,18,36)(11,21,20,26)(13,28,29,38)(15,32,33,40)(16,34,19,37)(22,24,30,35)(23,31,39,27)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399723);