Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,3,7,11,15,16,14,5,9,12,13,4,8,10,6)(17,18,21,26,22,25,36,40)(19,27,30,41,29,23,34,38)(20,31,33,42,32,24,35,39)(28,43,45,48,44,37,46,47);(2,6)(3,10)(4,11)(7,8)(9,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,27)(26,41)(30,36)(31,43)(32,44)(33,46)(35,45)(38,40)(39,48)(42,47);(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,31)(26,42)(27,43)(29,44)(30,46)(33,36)(34,45)(38,48)(39,40)(41,47)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,f^c=fg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38822);