Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2102 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 210 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,19,22,33)(12,16)(13,17)(14,18)(15,21,28,26)(20,36,38,35)(23,30)(24,31)(25,32)(27,34)(29,37,40,39);(1,3)(2,5)(4,8)(6,7)(13,25)(15,28)(17,32)(21,26)(23,27)(29,40)(30,34)(37,39);(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,33)(18,28)(19,32)(21,25)(23,39)(24,38)(27,37)(30,35)(31,40)(34,36);(3,8)(5,7)(9,12,14,24)(10,16,18,31)(11,20,22,38)(13,27,25,23)(15,29,28,40)(17,34,32,30)(19,35,33,36)(21,39,26,37)Polycyclic presentation:
2^9|a^2=eh,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356724);