Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3470 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 3470 conjugacy classes of subgroups — too many to draw. Only its 258 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:
(1,2)(3,4)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,34,36,27)(21,28,38,31)(23,29)(24,30)(25,32)(26,33)(35,39,37,40);(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,31)(18,19)(23,35)(24,36)(25,37)(26,38)(28,33)(29,40)(30,34)(32,39);(2,4)(6,8)(9,12,14,24)(10,16,18,30)(11,20,22,36)(13,25,26,23)(15,27,19,34)(17,32,33,29)(21,37,38,35)(28,40,31,39);(9,13,14,26)(10,17,18,33)(11,21,22,38)(12,23,24,25)(15,28,19,31)(16,29,30,32)(20,35,36,37)(27,39,34,40)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210036, #72227712, #72227720, … |
| maximal normal subgroup of | #72210036, #72227712, #72227720, … |
| maximal quotient of | #72240907, #72249442, C2×#449480 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356676);