Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| 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 |
| 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 | |
377 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 377 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)(3,6)(4,5)(7,8)(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);(2,5)(6,8)(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);(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);(1,3,4,7)(2,6,5,8)(11,20)(16,29)(19,36)(21,35)(25,38)(28,39)(30,31)(33,40)Polycyclic presentation:
2^9|b^2=i,d^2=ei,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^c=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232029, #72234321, #72234327, … |
| maximal normal subgroup of | #72232029, #72234321, #72234327, … |
| maximal quotient of | #72321065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420205);