Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
868 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,26)(14,17)(15,21)(18,29)(20,36)(22,35)(23,39)(24,40)(25,33)(27,31)(28,34)(30,37)(32,38);(1,2,5,7)(3,6,4,8)(9,11,14,22)(10,15,17,31)(12,23,25,20)(13,27,29,21)(16,32,33,30)(18,35,26,19)(24,37,28,38)(34,36,40,39);(3,4)(6,8)(9,12,14,25)(10,16,17,33)(11,20,22,23)(13,28,29,24)(15,30,31,32)(18,34,26,40)(19,36,35,39)(21,38,27,37);(1,3,5,4)(2,6,7,8)(10,17)(15,31)(16,33)(18,26)(19,35)(30,32)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284146, #72284163, #72284251 |
| maximal normal subgroup of | #72284146, #72284163, #72284251 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395253);