Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483463 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#41 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 240 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22,23,28)(19,25)(20,27)(24,31,30,29)(26,32);(17,18,19,22,20,24,25,29,21,23,26,28,27,30,32,31);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,23)(22,28)(24,30)(29,31);(3,9)(6,12)(10,15)(13,16)(18,23)(22,28)(24,30)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,f^b=fi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216671, #72216897, #72216945, … |
| maximal normal subgroup of | #72216671, #72216897, #72216945, … |
| maximal quotient of | #72285995, #72286277, #72343727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390662);