Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10,14,19,16,12,15,28)(11,20,24,37,25,38,23,36)(13,29,30,27,21,39,31,18)(17,32,34,35,26,40,33,22);(1,3,5,8,4,2,6,7)(9,11,21,35,14,24,31,40,16,25,13,22,15,23,30,32)(10,17,18,20,28,33,39,36,12,26,27,38,19,34,29,37);(2,3)(7,8)(9,12)(10,16)(11,17)(13,18)(14,19)(15,28)(20,35)(21,27)(22,38)(23,34)(24,33)(25,26)(29,31)(30,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44852);