Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152116 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,14,15,24)(4,20,7,12)(5,23)(6,11)(8,13)(9,16,25,31)(10,22,27,21)(17,29,32,26)(18,28,19,30);(1,3,8,19)(2,9,13,26)(4,18,7,15)(5,17,6,16)(10,14,27,30)(11,28,23,24)(12,29,20,25)(21,31,22,32);(1,4,8,7)(2,10)(3,15,19,18)(5,21,6,22)(9,14)(11,20)(12,23)(13,27)(16,31,17,32)(24,29)(25,28)(26,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306760, #72306778, #72306985 |
| maximal normal subgroup of | #72306760, #72306778, #72306985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59314);