Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485056 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
985 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 985 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,15)(4,20)(5,24)(6,10)(7,14)(8,17)(9,21)(11,16)(12,19)(13,27)(18,28)(22,31)(23,29)(25,32)(26,30);(1,3,6,18)(2,8,10,29)(4,21,22,26)(5,23,16,17)(7,19,27,25)(9,32,30,12)(11,28,24,15)(13,20,14,31);(1,4,7,23)(2,9,14,11)(3,16,19,21)(5,25,26,18)(6,22,27,17)(8,28,31,32)(10,30,13,24)(12,29,15,20);(2,10)(8,29)(9,30)(11,24)(12,32)(13,14)(15,28)(20,31)Polycyclic presentation:
2^9|b^2=g,c^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,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72324327, #72324337 |
| maximal normal subgroup of | #72324327, #72324337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392291);