Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | C2×#41 | M16 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,21,27,22,24,30,31,23,19,25,28,20,26,29,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(18,24,19,26)(21,25)(27,32,28,31)(29,30);(3,10)(6,14)(9,15)(11,16)(17,19,21,28,22,26,30,32,23,18,25,27,20,24,29,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cgh,e^b=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72287085, #72287197, #72287201, … |
| maximal normal subgroup of | #72287085, #72287197, #72287201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35851);