Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1782 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,32,27)(17,26,34,28)(23,36)(24,35)(29,33)(30,38)(31,37,40,39);(9,11)(10,16)(12,22)(13,27)(14,23)(15,24)(17,30)(18,35)(19,31)(20,32)(21,25)(26,37)(28,39)(29,38)(33,40)(34,36);(9,12,15,13)(10,17)(11,22,24,27)(14,26,29,28)(16,30)(18,33)(19,25)(20,34)(21,31)(23,37,38,39)(32,36)(35,40)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302708, #72302765, #72302787 |
| maximal normal subgroup of | #72302708, #72302765, #72302787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36490);