Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D4×D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×D4 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10,15,20)(11,21,25,52)(12,26,31,50)(13,32,35,66)(14,36,34,19)(16,38,41,22)(17,42,45,29)(18,46,48,28)(23,54,55,71)(24,56,37,51)(27,47,61,72)(30,64,63,58)(33,43,67,70)(39,65,68,60)(40,53,49,59)(44,62,57,69);(1,3)(2,5)(4,8)(6,7);(9,11,15,25)(10,16,20,41)(12,27,31,61)(13,33,35,67)(14,37,34,24)(17,43,45,70)(18,47,48,72)(19,49,36,40)(21,50,52,26)(22,42,38,29)(23,46,55,28)(30,65,63,60)(32,39,66,68)(44,71,57,54)(51,58,56,64)(53,62,59,69);(3,8)(5,7)Polycyclic presentation:
2^9|a^2=ei,c^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=de,f^c=fg,g^c=gh,h^c=hi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420396);