Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150567 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,18,14,16,25,33)(11,20,22,35,21,31,37,40)(13,19,28,36,29,17,26,34)(15,24,30,27,32,38,39,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,27)(14,21)(16,32)(17,31)(18,35)(19,20)(22,28)(23,29)(25,38)(26,37)(30,34)(33,40)(36,39);(2,6)(5,7)(10,16)(11,21)(12,25)(17,19)(23,27)(26,28)(30,39)(35,40)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57769);