Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
707 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 707 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,6,10,3,11,5,9,12,15,13,16,14,8)(17,18,20,23,30,27,31,32,21,24,29,28,22,19,25,26);(2,7)(3,12)(4,14)(6,13)(8,10)(9,15)(11,16)(17,19)(18,22)(20,28)(21,27)(23,29)(24,30)(25,26)(31,32);(20,29)(23,28)(25,31)(26,32)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=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,f^c=fgi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285149, #72285170, #72285321 |
| maximal normal subgroup of | #72285149, #72285170, #72285321 |
| maximal quotient of | #72287335, #72287351, #72287365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38827);