Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,14,21,15,22,32,20)(11,23,26,34,27,38,33,39)(12,19,28,18,29,16,13,30)(17,24,36,25,37,31,35,40);(1,2,5,7)(3,6,4,8)(10,16,22,19)(12,28)(13,29)(14,32)(17,34,37,39)(18,20,30,21)(23,35,38,36)(24,25)(26,33)(31,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17,22,37)(12,25)(13,31)(14,33)(15,27)(16,34,19,39)(18,38,30,23)(20,36,21,35)(24,28)(26,32)(29,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=fgi,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72307562, #72307565, #72307570, … |
| maximal normal subgroup of | #72307562, #72307565, #72307570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33638);