Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,19)(11,23)(12,20,29,37)(14,31,30,18)(15,22,16,21)(17,32)(24,38)(25,36)(26,35)(27,34)(28,39)(33,40);(2,5,7,6)(9,11,14,26,16,27,29,32,13,28,30,40,15,24,12,25)(10,17,20,34,22,35,18,23,19,36,37,38,21,33,31,39);(11,24,28,27)(12,29)(14,30)(17,33,36,35)(18,31)(20,37)(23,38,39,34)(25,26,32,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309441, #72309446, #72309450, … |
| maximal normal subgroup of | #72309441, #72309446, #72309450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35769);