Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126351 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 308 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:
(1,2,4,7)(3,5,8,6)(9,10,13,29,14,18,30,19)(11,20,23,34,21,15,31,38)(12,24,28,17,27,16,26,35)(22,32,40,37,25,36,39,33);(2,5)(3,8)(6,7)(9,11)(10,15,18,20)(12,25)(13,23)(14,21)(16,32,24,36)(17,33,35,37)(19,34,29,38)(22,27)(26,40)(28,39)(30,31);(10,16)(12,26)(13,30)(15,32)(17,29)(18,24)(19,35)(20,36)(22,39)(23,31)(25,40)(27,28)(33,38)(34,37)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=f,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72307554, #72307558, #72307573, … |
| maximal normal subgroup of | #72307554, #72307558, #72307573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33677);