Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #331 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | C4×#331 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 32 points:
(1,2,5,11,8,13,7,23)(3,14,17,26,19,27,16,9)(4,10,20,28,21,12,6,22)(15,29,31,25,32,30,18,24);(1,3,6,18,7,16,21,32,8,19,20,31,5,17,4,15)(2,9,12,25,11,14,22,30,13,26,10,24,23,27,28,29);(1,4,5,20,8,21,7,6)(2,10,11,28,13,12,23,22)(3,15,17,31,19,32,16,18)(9,24,14,29,26,25,27,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,d^b=dh,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225389, #72225390, #72225479, … |
| maximal normal subgroup of | #72225389, #72225390, #72225479, … |
| maximal quotient of | #72225520 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36093);