Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(2,5)(6,8)(9,10)(11,19,22,15)(12,23,26,16)(13,17)(14,18)(20,33)(21,34,36,29)(24,28)(25,30,38,31)(27,32)(35,39)(37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,24)(13,21)(14,22)(16,28)(17,29)(18,19)(20,26)(23,33)(25,37)(27,36)(30,39)(31,40)(32,34)(35,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20,22,24)(13,25)(14,26)(15,28,19,33)(17,31)(18,23)(21,35,36,37)(27,38)(29,40,34,39)(30,32);(3,7)(6,8)(9,13,14,27)(10,17,18,32)(11,21,22,36)(12,25,26,38)(15,29,19,34)(16,30,23,31)(20,35,24,37)(28,39,33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=ef,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381380);