Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152572 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13041 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #33559 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,3,7,9,21,12)(4,14,16,11,18,24,31,30)(6,10,23,28,22,27,20,13)(8,19,25,17,26,32,29,15);(1,3,5,2,7,12,21,9)(4,11,19,8,18,30,32,26)(6,13,20,10,22,28,23,27)(14,17,29,31,24,15,25,16);(1,4)(2,8)(3,11)(5,19)(6,15)(7,18)(9,26)(10,14)(12,30)(13,25)(16,20)(17,22)(21,32)(23,31)(24,27)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59769);