Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2194 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,17)(14,18)(19,28)(20,26)(21,27)(22,35)(23,30)(24,31)(25,33)(29,36)(32,37)(34,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,28)(17,26)(18,27)(22,32)(23,33)(24,34)(25,30)(29,39)(31,38)(35,37)(36,40);(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,29)(18,30)(20,32)(21,33)(24,36)(26,37)(27,28)(31,35)(34,40)(38,39);(3,8)(5,7)(9,13,14,24)(10,17,18,31)(11,20,21,34)(12,22,23,36)(15,26,27,38)(16,29,30,35)(19,32,33,40)(25,37,28,39)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210036, #72210037, #72227716, … |
| maximal normal subgroup of | #72210036, #72210037, #72227716, … |
| maximal quotient of | #72240908, #72240916, #72240919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325093);