Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486115 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,16)(4,13)(5,10)(6,25)(7,14)(8,26)(9,15)(11,17)(12,21)(18,28)(19,30)(20,27)(22,32)(23,31)(24,29);(2,10)(4,21)(5,17)(6,22)(11,14)(12,30)(13,27)(15,29)(16,25)(23,24)(26,31)(28,32);(1,3)(2,11)(4,17)(5,21)(6,23)(7,18)(8,19)(9,20)(10,14)(12,30)(13,27)(15,31)(16,25)(22,24)(26,29)(28,32);(1,4,9,6)(2,12,15,32)(3,17,20,23)(5,19,24,18)(7,21,8,22)(10,27,29,16)(11,30,31,28)(13,26,25,14)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72278749, #72338756, #72338759, … |
| maximal normal subgroup of | #72278749, #72338756, #72338759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393312);