Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124851 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
281 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 281 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10,13,16,14,20,28,18,15,19,29,32,30,37,12,26)(11,21,23,33,24,17,22,31,25,35,39,36,40,34,27,38);(1,2)(3,6)(4,8)(5,7)(10,16,19,32)(12,13)(14,30)(17,31,34,38)(18,20,26,37)(21,36,35,33)(22,39)(23,27)(24,40)(28,29);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,17)(12,27,28,22)(13,23,29,39)(14,24,30,40)(16,31)(18,36)(19,34)(20,35)(21,37)(26,33)(32,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fgh,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72307348, #72307353, #72307363, … |
| maximal normal subgroup of | #72307348, #72307353, #72307363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32118);