Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23904 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 731 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:
(1,2,6,11,7,10,22,26)(3,12,15,5,9,13,21,20)(4,16,18,27,19,8,23,24)(14,29,30,32,25,28,31,17);(1,3,13,11,7,9,12,26)(2,6,21,5,10,22,15,20)(4,14,29,27,19,25,28,24)(8,18,30,17,16,23,31,32);(1,4)(2,8)(3,14)(5,17)(6,18)(7,19)(9,25)(10,16)(11,27)(12,28)(13,29)(15,31)(20,32)(21,30)(22,23)(24,26)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72258290 |
| maximal normal subgroup of | #72258290 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46953);