Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245147 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1682 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1682 conjugacy classes of subgroups — too many to draw. Only its 156 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:
(1,2,5,10)(3,12,13,29)(4,16,19,28)(6,11,20,9)(7,21,23,14)(8,25,27,17)(15,26,31,30)(18,24,32,22);(4,17)(7,22)(8,16)(12,29)(14,18)(19,25)(21,32)(23,24)(26,30)(27,28);(1,3,6,15)(2,7,11,24)(4,18,17,14)(5,13,20,31)(8,26,16,12)(9,22,10,23)(19,32,25,21)(27,30,28,29);(4,19)(8,27)(12,29)(14,21)(16,28)(17,25)(18,32)(26,30)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227597, #72227709, #72341001, … |
| maximal normal subgroup of | #72227597, #72227709, #72341001, … |
| maximal quotient of | #72209840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152354);