Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23913 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,6,10,7,12,4,18)(3,13,16,23,17,29,14,8)(5,9,19,24,20,11,21,26)(15,27,30,25,31,28,32,22);(1,3)(2,8,12,23)(4,16,6,14)(5,15)(7,17)(9,22,11,25)(10,13)(18,29)(19,32,21,30)(20,31)(24,27)(26,28);(2,9)(4,6)(5,20)(8,22)(10,24)(11,12)(13,27)(14,16)(15,31)(18,26)(23,25)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72302901, #72302905, #72302915 |
| maximal normal subgroup of | #72302901, #72302905, #72302915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46930);