Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #47179 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,14,19,15,18,13,27)(11,20,22,31,23,16,28,32)(12,24,25,17,26,33,29,34)(21,36,38,30,39,40,37,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,20)(19,31)(24,40)(25,38)(26,39)(27,32)(29,37)(33,36)(34,35);(9,12,15,26)(10,17,18,34)(11,21,23,39)(13,29,14,25)(16,30,20,35)(19,33,27,24)(22,38,28,37)(31,36,32,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,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, 37319);