Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1409 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7403 |
| prime radical | #532 | D4 |
| LCS | G ▷ #376 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,11,17,28)(3,12,13,26,15,23,30,32)(6,20,21,29,19,10,18,27)(7,16,22,31,8,24,25,14);(2,7,11,8)(9,26,28,32)(10,12,20,23)(13,30)(16,24)(22,27,25,29)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,d^c=dghi,e^c=ei,e^d=ehi,f^d=fi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72339749, #72339795 |
| maximal normal subgroup of | #72339749, #72339795 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1843);