Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9061 |
| prime radical | #19616 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,7,14,8,15,28,31,9,13,27,32,6,17,5,12)(3,16,20,4,21,24,18,25,19,22,26,10,11,29,30,23);(1,3,8,11,9,19,6,21)(2,10,15,25,13,4,17,23)(5,26,7,18,28,20,27,30)(12,24,14,16,31,29,32,22);(1,4)(2,11)(3,17)(5,24)(6,25)(7,16)(8,23)(9,10)(12,30)(13,21)(14,26)(15,19)(18,31)(20,32)(22,27)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214466, #72214690, #72214702, … |
| maximal normal subgroup of | #72214466, #72214690, #72214702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44751);