Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480662 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 40 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19)(12,16)(13,26,27,17)(14,18)(15,22,28,35)(20,33)(21,34)(23,37)(24,39,40,31)(25,32)(29,36,30,38);(2,6)(3,4)(9,11)(10,15)(12,23)(13,22)(14,20)(16,30)(17,19)(18,28)(21,25)(24,38)(26,33)(27,35)(29,32)(31,37)(34,39)(36,40);(1,3,5,4)(2,7,6,8)(11,20)(15,28)(19,33)(21,23)(22,35)(29,30)(34,37)(36,38);(2,6)(7,8)(9,12,14,25)(10,16,18,32)(11,21,20,23)(13,24,27,40)(15,29,28,30)(17,31,26,39)(19,34,33,37)(22,36,35,38)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fg,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284174, #72284184, #72284199, … |
| maximal normal subgroup of | #72284174, #72284184, #72284199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387914);