Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2635 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,5,6,7)(9,10)(11,20,23,36)(12,24,28,34)(13,18,30,17)(14,19)(15,22)(16,27)(21,33)(25,31)(26,35)(29,32)(37,38,40,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15,19,32)(12,25,28,21)(13,29,30,22)(16,33,35,31)(17,36,18,20)(24,37,34,40)(26,38,27,39);(9,12)(10,16,19,35)(11,21)(13,26)(14,28)(15,31,32,33)(17,24,18,34)(20,37,36,40)(22,38)(23,25)(27,30)(29,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38466);