Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,16)(4,21)(5,25)(6,13)(7,11)(8,15)(9,19)(10,22)(12,18)(14,17)(20,32)(23,30)(24,29)(26,28)(27,31);(1,3,17,26)(2,9,14,21)(4,18,19,27)(5,24,22,23)(6,7,20,8)(10,28,25,16)(11,29,15,30)(12,32,31,13);(1,4,8,24)(2,10,15,12)(3,18,6,22)(5,26,27,20)(7,23,17,19)(9,28,30,32)(11,31,14,25)(13,21,16,29);(2,11)(9,29)(10,31)(12,25)(13,28)(14,15)(16,32)(21,30)Polycyclic presentation:
2^9|b^2=gi,c^2=i,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^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72232556, #72232570, #72234710, … |
| maximal normal subgroup of | #72232556, #72232570, #72234710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392275);