Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136630 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1875 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8943 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 32 points:
(1,2,6,11,7,4,18,10)(3,12,17,30)(5,9,21,28,13,19,31,27)(8,23,14,20)(15,24,32,25)(16,26,22,29);(1,3,13,23,18,16,21,15,7,17,5,20,6,22,31,32)(2,8,19,29,10,25,28,12,4,14,9,26,11,24,27,30);(1,4)(2,7)(3,14)(5,9)(6,10)(8,17)(11,18)(12,15)(13,19)(16,24)(20,26)(21,28)(22,25)(23,29)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43826);