Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2084 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2084 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,21,28)(12,16,23,31)(13,24,25,17)(19,26,33,29)(20,34,35,27)(22,30,36,37)(32,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,28)(12,19,23,33)(13,20,25,35)(16,29,31,26)(17,27,24,34)(22,32,36,40)(30,39,37,38);(5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,30)(18,31)(20,32)(21,33)(24,37)(25,36)(27,38)(28,29)(34,39)(35,40);(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,30)(18,24)(19,32)(21,35)(23,36)(26,38)(28,34)(29,39)(31,37)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227298, #72227718, #72227719, … |
| maximal normal subgroup of | #72227298, #72227718, #72227719, … |
| maximal quotient of | #72267910, #72268074, #72268080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152444);