Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141233 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,5,9,6,10,4,7)(3,11,13,16,14,15,12,8)(17,18,21,25,22,26,30,31,20,24,19,27,29,32,28,23);(2,7,10,9)(8,15,16,11)(18,23,24,31)(19,28)(21,30)(22,29)(25,26,27,32);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(18,24)(23,31)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72285422, #72285431, #72285524, … |
| maximal normal subgroup of | #72285422, #72285431, #72285524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48525);