Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21750 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60817 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
908 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 32 points:
(1,2)(3,12,17,29)(4,18,20,8)(5,10)(6,11)(7,15,25,13)(9,19,28,32)(14,22)(16,23)(21,27)(24,30)(26,31);(1,3,6,17)(2,7,11,25)(4,14,20,16)(5,13,21,15)(8,26,18,24)(9,22,28,23)(10,29,27,12)(19,30,32,31);(2,8)(3,13)(5,21)(7,22)(9,27)(10,28)(11,18)(12,24)(14,30)(15,17)(16,31)(19,32)(23,25)(26,29);(1,4,6,20)(2,9)(3,14,17,16)(5,19,21,32)(7,23)(8,27)(10,18)(11,28)(12,24)(13,30,15,31)(22,25)(26,29)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gh,f^d=fi,g^d=gi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403447);