Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 550 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,18,13,19,14,28)(11,20,29,31,23,36,21,16)(12,17,24,32,26,34,27,33)(22,35,37,30,25,39,38,40);(1,3,4,7)(2,5,6,8)(11,21)(12,24)(14,15)(16,20)(17,32)(18,28)(22,25)(23,29)(26,27)(31,36)(33,34)(35,39);(2,6)(5,8)(9,11,14,21,13,23,15,29)(10,16,18,20,19,31,28,36)(12,25,24,38,26,22,27,37)(17,30,33,35,34,40,32,39);(3,7)(5,8)(9,12,13,26)(10,17)(11,22,23,25)(14,24,15,27)(16,30)(18,33)(19,34)(20,35)(21,37,29,38)(28,32)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=ef,b^2=efgh,c^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387084);