Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 512 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,8,7)(9,10)(11,15)(12,16)(13,26)(14,27)(17,20)(18,21)(19,25)(22,28)(23,34)(24,40)(29,37)(30,36)(31,35)(32,38)(33,39);(5,7,8,6)(10,15)(11,20)(13,21)(14,25)(16,28)(17,18)(19,34)(22,35)(23,39)(24,38)(26,29)(27,30)(31,32)(33,40)(36,37);(1,2,3,4)(10,16)(15,28)(17,31)(18,32)(19,27)(26,40)(29,33)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ei,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60060);