Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487643 |
| 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×C22⋊C4 | C24 |
| prime radical | #57964 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 977 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:
(1,2,7,13)(3,15,18,31)(4,20,17,10)(5,24,21,12)(6,11,19,30)(8,26,9,14)(16,27,25,32)(22,28,23,29);(1,3,7,18)(2,10,13,20)(4,21,17,5)(6,8,19,9)(11,27,30,32)(12,31,24,15)(14,28,26,29)(16,22,25,23);(2,11)(4,22)(5,16)(10,27)(12,28)(13,30)(14,15)(17,23)(20,32)(21,25)(24,29)(26,31);(1,4,8,22)(2,10,26,29)(3,16,19,5)(6,21,18,25)(7,17,9,23)(11,24,31,27)(12,15,32,30)(13,20,14,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72234734, #72234736, #72234738 |
| maximal normal subgroup of | #72234734, #72234736, #72234738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395347);