Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483032 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#331 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #60646 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1077 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 1077 conjugacy classes of subgroups — too many to draw. Only its 312 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:
(1,2,4,10,16,7,15,20,6,11,18,25,5,9,17,24)(3,8,13,23,27,19,28,30,12,21,26,31,14,22,29,32);(2,7)(4,15)(5,16)(8,19)(9,11)(10,25)(13,28)(14,27)(17,18)(21,22)(23,31)(26,29);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,21)(15,28)(16,27)(17,29)(18,26)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,22)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=f,e^2=h,f^2=ghi,g^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215816, #72219676, #72219679 |
| maximal normal subgroup of | #72215816, #72219676, #72219679 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390232);