Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485315 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #18162 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1158 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 120 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)(4,7)(5,9)(6,11)(8,14)(10,16)(13,26)(15,27)(17,20)(18,21)(19,22)(23,31)(24,32)(25,30)(28,29);(2,7)(4,16)(8,20)(9,10)(11,21)(12,26)(13,29)(17,32)(18,30)(22,25)(23,24)(27,28);(1,3,5,15)(2,8,22,27)(4,17,16,32)(6,14,19,31)(7,20,25,28)(9,23,11,12)(10,24,21,26)(13,30,29,18);(2,9)(7,10)(8,23)(11,22)(12,27)(20,24)(21,25)(26,28)Polycyclic presentation:
2^9|c^2=ghi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231386, #72231851, #72231952, … |
| maximal normal subgroup of | #72231386, #72231851, #72234737, … |
| maximal quotient of | #72237728, #72237836, #72237856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392581);