Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1955 | C22 |
| socle | C22 | #1352 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1955 ◁ G | |
| LECS | G ▷ #1955 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
161 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 32 points:
(1,2)(3,10,12,6)(4,8,11,7)(5,9)(13,14,16,15)(18,22)(19,26,29,30)(23,32)(24,27)(25,31);(1,3,11,16,5,13,4,12)(2,6,8,14,9,15,7,10)(17,18,20,24,21,25,28,23)(19,27,26,22,29,32,30,31)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=fghi,e^2=hi,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defgi,e^b=ei,g^b=gi,d^c=dghi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 437);