Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395181 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1232 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,20)(12,24,26,33)(13,27)(14,18)(15,19)(16,22)(17,25,34,28)(21,30,37,40)(23,31)(29,32)(35,38,39,36);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,27,31)(12,21,28,38,26,37,25,36)(17,30,24,35,34,40,33,39);(3,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,27)(15,19)(20,22)(21,30)(23,32)(24,28)(25,33)(26,34)(29,31)(35,38)(36,39)(37,40);(9,12,13,28,15,26,14,25)(10,17,18,24,19,34,27,33)(11,21,22,38,23,37,29,36)(16,30,20,35,32,40,31,39)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310310);