Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
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 | #1767 |
| prime radical | C23 | #432 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(2,6,8,11)(7,13,14,15)(18,23,25,31)(19,27)(21,30)(24,32,26,28);(1,2)(3,7)(4,11)(5,8)(6,12)(9,15)(10,14)(13,16)(17,18,21,26,20,23,27,32,22,25,30,24,29,31,19,28);(1,3,4,9,5,10,12,16)(2,7,11,15,8,14,6,13)(17,19,20,21,22,27,29,30)(18,24,31,32,25,26,23,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ci,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 | #72285417, #72285422, #72285438, … |
| maximal normal subgroup of | #72285417, #72285422, #72285438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36275);