Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130089 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,21)(12,26,13,18)(14,19)(15,20)(16,23,33,22)(17,27,35,28)(24,31)(25,36)(29,34)(30,37)(32,40,38,39);(2,6)(3,4)(7,8)(9,11,15,25)(10,16,20,33)(12,23,13,22)(14,29,30,24)(17,34,35,31)(18,36,26,21)(19,32,37,38)(27,39,28,40);(9,12,15,13)(10,17,20,35)(11,22,25,23)(14,28,30,27)(16,31,33,34)(18,37,26,19)(21,38,36,32)(24,40,29,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37427);