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 | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
648 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2,4,7,5,8,3,6)(9,10)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,32,26)(17,27,35,28)(23,33)(24,36)(29,34)(30,38)(31,37,40,39);(9,11,15,24)(10,16)(12,26,13,22)(14,23,29,38)(17,33)(18,21)(19,31)(20,32)(25,36)(27,39,28,37)(30,35)(34,40);(1,3,5,4)(2,6,8,7)(9,12,15,13)(10,17)(11,22,24,26)(14,27,29,28)(16,30)(18,34)(19,25)(20,35)(21,31)(23,37,38,39)(32,33)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fgi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302708, #72302772, #72302787 |
| maximal normal subgroup of | #72302708, #72302772, #72302787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36590);