Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301973 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62269 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1142 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,17,23,32)(12,24,28,39)(13,29,14,19)(16,30,33,22)(18,27,36,26)(21,37,25,31)(34,40,35,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,30,15,23,13,22)(10,16,29,32,20,33,19,17)(12,21,27,40,28,25,26,38)(18,31,39,35,36,37,24,34);(1,2)(3,7,8,5)(4,6)(9,11)(10,17)(12,25)(13,22)(14,30)(15,23)(16,29)(18,35)(19,33)(20,32)(21,28)(24,31)(26,40)(27,38)(34,36)(37,39);(9,12,13,26,15,28,14,27)(10,18,19,24,20,36,29,39)(11,21,22,38,23,25,30,40)(16,31,17,34,33,37,32,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209172);