Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152598 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13047 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3147 | C22 |
| 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 | |
476 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 476 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:
(1,2,7,12,8,11,27,30)(3,14,17,4,16,21,28,22)(5,23,25,29,26,10,6,13)(9,18,19,15,20,32,31,24);(1,3,6,18,8,16,25,32)(2,9,29,22,11,20,13,4)(5,24,27,17,26,15,7,28)(10,21,30,19,23,14,12,31);(1,4)(2,3)(5,14)(6,9)(7,19)(8,22)(10,17)(11,16)(12,24)(13,32)(15,30)(18,29)(20,25)(21,26)(23,28)(27,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eghi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283528, #72283558, #72283655, … |
| maximal normal subgroup of | #72283528, #72283558, #72283655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59797);