Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C23 | #446 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,13)(8,14,15,16)(17,18,21,26,20,25,29,32,22,27,30,19,24,23,31,28);(1,2,5,7,6,9,4,13)(3,8,11,14,12,15,10,16)(18,23,27,25)(19,28,26,32)(21,30)(29,31);(1,3,4,10,6,12,5,11)(2,8,13,16,9,15,7,14)(17,19,29,18,24,32,21,23,22,26,31,27,20,28,30,25)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287085, #72287175, #72287184, … |
| maximal normal subgroup of | #72287085, #72287175, #72287184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36390);