Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #265283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1061 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1061 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,13,31)(12,27,30,36)(14,21,15,20)(17,26,19,25)(18,32,35,29)(24,34,28,33)(37,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11,15,26,16,13,14,25)(10,17,21,31,22,19,20,23)(12,24,32,39,30,28,29,38)(18,33,36,40,35,34,27,37);(5,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,32)(16,30)(17,33)(19,34)(20,27)(21,36)(22,35)(23,37)(25,38)(26,39)(31,40);(9,13,15,25,16,11,14,26)(10,19,21,23,22,17,20,31)(12,28,32,38,30,24,29,39)(18,34,36,37,35,33,27,40)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172483);