Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1680 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,5,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,34)(23,36)(24,30)(25,31)(29,37)(32,40)(33,39)(35,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,28)(12,22)(13,20,25,33)(16,26)(17,27,31,39)(19,24)(23,35)(29,38)(30,34)(32,37)(36,40);(5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,30)(20,32)(21,22)(25,37)(27,38)(28,34)(31,36)(33,35)(39,40);(9,13)(10,17)(11,20)(12,23)(14,25)(15,27)(16,29)(18,31)(19,32)(21,33)(22,35)(24,37)(26,38)(28,39)(30,36)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=eh,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227298, #72241004, #72252238, … |
| maximal normal subgroup of | #72227298, #72241004, #72252238, … |
| maximal quotient of | #72266204, #72266207, #72266210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159045);