Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125887 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1659 |
| commutator, Frattini, cosocle | #376 | C23 |
| socle | C2 | #7755 |
| prime radical | #532 | D4 |
| LCS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #376 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(2,8)(3,12,14,15)(9,21,25,32)(10,23)(11,22)(13,16,30,28)(20,31)(24,26,29,27);(1,2)(3,9)(4,8)(5,11)(6,20)(7,10)(12,26)(13,29)(14,25)(15,27)(16,21)(17,31)(18,23)(19,22)(24,30)(28,32);(1,3,6,12,7,14,5,15)(2,9,20,26,10,25,11,27)(4,16,17,13,18,28,19,30)(8,21,31,29,23,32,22,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgh,d^2=f,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72339743, #72339806 |
| maximal normal subgroup of | #72339743, #72339806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33153);