Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60741 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1102 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,7,8,5)(4,6)(9,10,14,18,15,19,13,20)(11,21,23,16,24,33,30,32)(12,25,27,35,29,17,28,36)(22,37,26,31,39,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,33)(19,32)(20,21)(22,28)(25,37)(27,39)(29,38)(31,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,31)(18,35)(19,25)(20,36)(21,34)(23,38)(24,39)(26,30)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fh,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18682);