Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,7)(4,6)(5,8)(10,15)(11,19)(13,24)(14,26)(16,30)(17,32)(18,23)(20,36)(21,33)(22,37)(25,28)(27,34)(29,31)(35,40)(38,39);(1,3,4,8)(2,5,6,7)(9,10,12,15)(11,16,20,29)(13,17,22,27)(14,18,23,26)(19,31,36,30)(21,28,35,39)(24,34,37,32)(25,33,38,40);(3,8)(5,7)(9,11)(10,16)(12,20)(13,25)(14,28)(15,29)(17,33)(18,35)(19,24)(21,26)(22,38)(23,39)(27,40)(30,32)(31,34)(36,37);(2,6)(5,7)(9,10,12,15)(11,16,20,29)(13,17,22,27)(14,18,23,26)(19,31,36,30)(21,28,35,39)(24,34,37,32)(25,33,38,40)Polycyclic presentation:
2^9|b^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=cei,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420310);