Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1722 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 562 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) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,13,12)(11,15,16,14)(27,33)(28,35)(30,36)(31,38);(2,5)(6,8)(9,11,13,16)(10,14,12,15)(18,22)(23,24)(27,33)(30,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(17,19,21,20)(18,23,22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,24,22,23)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863261);