Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245084 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1309 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1309 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,27)(21,28)(22,36)(23,31)(24,32)(25,33)(29,35)(30,38)(34,40)(37,39);(9,11,14,21)(10,15,18,28)(12,19,23,35)(13,24,25,20)(16,29,31,26)(17,32,33,27)(22,37,38,34)(30,39,36,40);(2,5)(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,30)(18,31)(20,34)(21,35)(24,37)(25,38)(27,39)(28,29)(32,40)(33,36);(9,13,14,25)(10,17,18,33)(11,20,21,24)(12,22,23,38)(15,27,28,32)(16,30,31,36)(19,34,35,37)(26,39,29,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152286);