Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1333 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7178 |
| prime radical | C23 | #362 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,7,15,24,48,9,16,32,14,8,36,6,33)(3,17,19,52,21,54,30,53,22,39,34,62,35,44,59,64)(4,23,25,13,27,49,26,45,29,46,31,50,28,47,58,11)(10,37,38,18,41,55,51,20,43,60,63,57,42,56,40,61);(1,3,4,20,6,34,26,60,8,21,28,57,32,59,25,56,9,22,29,61,24,19,58,37,7,35,27,18,5,30,31,55)(2,10,13,39,12,40,47,64,15,42,45,54,48,63,23,62,16,43,50,17,14,51,49,53,36,41,11,44,33,38,46,52)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fgh,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72223659, #72223669, #72223683, … |
| maximal normal subgroup of | #72223659, #72223669, #72223683, … |
| maximal quotient of | #72223658, C2×#94058 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1254);