Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1806 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 1806 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)(11,19,22,15)(12,23)(13,17)(14,18)(16,25)(20,27,34,31)(21,32,35,28)(24,36)(26,30)(29,37)(33,38,40,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,28)(18,19)(23,31)(24,33)(25,34)(26,35)(29,38)(30,32)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,27,19,31)(17,29,30,36)(21,33)(22,34)(26,37)(28,38,32,39)(35,40);(9,13,14,26)(10,17,18,30)(11,21,22,35)(12,24,25,37)(15,28,19,32)(16,29,23,36)(20,33,34,40)(27,38,31,39)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^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 | #72310819, #72310854, #72313188 |
| maximal normal subgroup of | #72310819, #72310854, #72313188 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97295);