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 | #47665 | 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 | |
664 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 664 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)(9,10,14,19,15,18,13,28)(11,20,22,32,23,16,29,33)(12,24,26,17,27,35,30,36)(21,37,39,31,25,40,38,34);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,25)(13,29)(14,22)(15,23)(17,34)(18,20)(19,32)(21,27)(24,37)(26,38)(28,33)(30,39)(31,36)(35,40);(9,12)(10,17)(11,21)(13,30)(14,26)(15,27)(16,31)(18,36)(19,35)(20,34)(22,39)(23,25)(24,28)(29,38)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,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, 37322);