Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94676 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1412 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7428 |
| 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 | |
141 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 141 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:
(1,2,3,10,7,11,17,28)(4,9,18,24,20,27,31,29)(5,12,15,25,6,13,16,30)(8,21,22,14,23,19,26,32);(1,3,5,15,7,17,6,16)(2,8,11,23)(4,18,19,32,20,31,21,14)(9,25)(10,26)(12,24,13,29)(22,28)(27,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=defghi,g^b=gi,d^c=dghi,e^c=ei,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1905);