Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
885 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 885 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:
(2,5)(3,7)(6,8)(10,15)(12,24)(13,16)(17,27)(19,29)(20,35)(21,22)(25,26)(28,34)(30,39)(31,32)(36,37);(9,10,16,29,14,19,13,15)(11,17,22,31,23,32,21,27)(12,18,24,34,26,33,25,28)(20,30,37,40,36,39,35,38);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,20)(13,21)(14,23)(15,27)(16,22)(18,30)(19,32)(24,37)(25,35)(26,36)(28,38)(29,31)(33,39)(34,40);(3,7)(5,8)(9,12)(10,18)(11,20)(13,25)(14,26)(15,28)(16,24)(17,30)(19,33)(21,35)(22,37)(23,36)(27,38)(29,34)(31,40)(32,39)Polycyclic presentation:
2^9|b^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255280);