Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5,6,7)(9,10)(11,16)(12,17)(13,18,26,15)(14,19)(20,29)(21,30,36,27)(22,31)(23,32)(24,33,38,28)(25,34)(35,40,37,39);(1,2)(3,7)(4,6)(5,8)(10,15,19,18)(16,27,31,30)(17,28,34,33)(29,39,32,40);(9,11,14,22)(10,16,19,31)(12,23,25,20)(13,21,26,36)(15,27,18,30)(17,32,34,29)(24,37,38,35)(28,40,33,39);(9,12,14,25)(10,17,19,34)(11,20,22,23)(13,24,26,38)(15,28,18,33)(16,29,31,32)(21,35,36,37)(27,39,30,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=i,d^2=i,e^2=hi,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^e=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388698);