Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131824 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61475 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
594 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 40 points:
(2,5,6,7)(9,10,13,20,15,22,14,21)(11,23,25,37,27,33,26,38)(12,28,31,18,32,16,29,19)(17,30,34,40,36,39,35,24);(1,2)(3,7)(4,6)(5,8)(10,16)(12,29)(13,14)(17,33)(18,20)(19,21)(22,28)(23,36)(24,30)(25,26)(31,32)(34,37)(35,38)(39,40);(9,11)(10,17)(12,30)(13,25)(14,26)(15,27)(16,33)(18,37)(19,38)(20,34)(21,35)(22,36)(23,28)(24,29)(31,40)(32,39)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39024);