Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4216 conjugacy classes of subgroups; only the 610 normal subgroups are shown.
This group has 4216 conjugacy classes of subgroups — too many to draw. Only its 610 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 — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,13)(11,16,22,20)(12,17,25,23)(14,26,27,18)(19,31,33,28)(21,34,35,29)(24,36,37,30)(32,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,28)(18,29)(23,31)(24,32)(25,33)(26,34)(27,35)(30,38)(36,39)(37,40);(3,7)(5,8)(9,12)(10,17)(11,19)(13,23)(14,24)(15,25)(16,28)(18,30)(20,31)(21,32)(22,33)(26,36)(27,37)(29,38)(34,39)(35,40);(2,6)(5,8)(9,13,15,10)(11,20,22,16)(12,23,25,17)(14,18,27,26)(19,31,33,28)(21,29,35,34)(24,30,37,36)(32,38,40,39);(9,14,15,27)(10,18,13,26)(11,21,22,35)(12,24,25,37)(16,29,20,34)(17,30,23,36)(19,32,33,40)(28,38,31,39)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340157);