Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1533 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | #3498 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 192 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:
(1,2,4,10,5,7,19,22,6,11,18,26,20,9,16,25)(3,12,14,27,15,23,30,32,13,8,17,24,28,21,29,31);(2,7)(4,16)(5,20)(8,21)(9,11)(10,26)(12,23)(14,29)(15,28)(17,30)(18,19)(24,27);(1,3)(2,8)(4,17)(5,15)(6,13)(7,21)(9,23)(10,27)(11,12)(14,18)(16,30)(19,29)(20,28)(22,32)(24,26)(25,31);(3,13)(8,12)(14,17)(15,28)(21,23)(24,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=f,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^c=fi,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390217);