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 | #3537 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1144 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1144 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)(9,10,15,21)(11,22,12,17)(13,28,30,36)(14,20,26,19)(16,25,32,24)(18,34,37,29)(23,38,27,31)(33,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,23,29,39,30,27,34,40)(18,31,28,35,37,38,36,33);(1,2)(3,7,8,5)(4,6)(9,12)(10,17)(11,15)(13,23)(14,25)(16,19)(18,35)(20,32)(21,22)(24,26)(27,30)(28,38)(29,39)(31,36)(33,37)(34,40);(9,13,14,29,15,30,26,34)(10,18,19,28,21,37,20,36)(11,23,24,39,12,27,25,40)(16,31,22,35,32,38,17,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,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, 209170);