Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149791 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2126 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12695 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 623 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)(3,14,17,29)(4,19,7,12)(5,22)(6,11)(8,13)(9,16,25,32)(10,21,27,20)(15,24,18,30)(23,28,31,26);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,18)(10,14)(11,24)(12,28)(13,26)(19,25)(20,32)(21,31)(22,29)(27,30);(1,4,8,7)(2,10)(3,15,18,17)(5,20,6,21)(9,14)(11,19)(12,22)(13,27)(16,31,23,32)(24,25)(26,30)(28,29)Polycyclic presentation:
2^9|a^2=f,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,d^b=dh,f^b=fghi,g^b=gi,f^c=fi,f^d=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59272);