Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150171 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | C2×#1343 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,13,15,29)(4,18,20,28)(5,11)(6,12)(7,9)(8,14,23,30)(10,21,26,22)(16,31,17,27)(19,24,32,25);(1,3)(2,8)(4,19)(5,17)(6,16)(7,15)(9,23)(10,27)(11,25)(12,24)(13,18)(14,21)(20,32)(22,30)(26,31)(28,29);(2,9)(8,23)(10,26)(11,12)(13,29)(18,28)(24,25)(27,31)Polycyclic presentation:
2^9|a^2=e,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338354, #72338357, #72338358, … |
| maximal normal subgroup of | #72338354, #72338357, #72338358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57367);