Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485605 |
| 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 | #46996 | 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 | |
1145 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1145 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:
(1,2,5,10)(3,8,12,22)(4,7,15,9)(6,11,17,24)(13,21,25,31)(14,18,26,20)(16,19,28,23)(27,29,32,30);(2,7)(3,12)(4,15)(8,20)(9,10)(11,19)(13,25)(16,28)(18,22)(21,30)(23,24)(29,31);(3,13)(4,16)(8,21)(12,25)(15,28)(18,29)(20,30)(22,31);(1,3)(2,8)(4,14)(5,12)(6,13)(7,18)(9,20)(10,22)(11,21)(15,26)(16,27)(17,25)(19,29)(23,30)(24,31)(28,32)Polycyclic presentation:
2^9|a^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72227772, #72227813, #72231805, … |
| maximal normal subgroup of | #72227772, #72231851, #72234734, … |
| maximal quotient of | #72237744, #72237865, #72284024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392881);