Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
757 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 757 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,5,7,8)(9,10,13,16,15,20,14,19)(11,21,23,17,25,32,24,34)(12,26,28,36,30,31,29,18)(22,33,27,37,39,40,38,35);(1,3)(2,5)(4,7)(6,8)(10,16,20,19)(17,32,34,21)(18,26,36,31)(33,37,40,35);(3,7)(5,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,32)(20,34)(22,28)(26,40)(29,39)(30,38)(31,33)(36,37);(9,12)(10,18,20,36)(11,22)(13,29)(14,28)(15,30)(16,31,19,26)(17,33,34,40)(21,37,32,35)(23,38)(24,27)(25,39)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386500);