Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1766 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 4 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,20,15,19,13,18)(11,21,23,34,24,16,29,31)(12,25,28,32,27,17,26,33)(22,30,36,39,37,35,38,40);(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,31)(19,21)(20,34)(25,35)(26,36)(27,37)(28,38)(32,39)(33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17,19,25)(11,22)(13,26)(14,28)(15,27)(16,30,21,35)(18,32,20,33)(23,36)(24,37)(29,38)(31,39,34,40);(9,13,15,14)(10,18,19,20)(11,23,24,29)(12,26,27,28)(16,31,21,34)(17,32,25,33)(22,36,37,38)(30,39,35,40)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72310774, #72310789 |
| maximal normal subgroup of | #72310774, #72310789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97326);