Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190116 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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)(20,28,36,33)(21,34,38,29)(24,31)(26,30)(27,32)(35,39,37,40);(9,11,14,22)(10,15,18,19)(12,20,25,36)(13,21,27,38)(16,28,23,33)(17,29,32,34)(24,35,26,37)(30,39,31,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16,18,23)(11,20)(13,26)(14,25)(15,28,19,33)(17,31,32,30)(21,37)(22,36)(24,27)(29,40,34,39)(35,38);(9,13,14,27)(10,17,18,32)(11,21,22,38)(12,24,25,26)(15,29,19,34)(16,30,23,31)(20,35,36,37)(28,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284639, #72284696, #72284804, … |
| maximal normal subgroup of | #72284639, #72284696, #72284804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97315);