Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10018 |
| commutator, Frattini, cosocle | C4×M16 | C23 |
| prime radical | #33688 | C2 |
| LCS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(2,8)(4,17)(5,6)(9,25)(10,18)(11,23)(12,24)(13,29)(14,15)(19,30)(20,21)(22,26)(27,32)(28,31);(1,2)(3,9)(4,18)(5,10)(6,11)(7,12)(8,17)(13,22)(14,26)(15,27)(16,28)(19,31)(20,23)(21,24)(25,29)(30,32);(1,3,7,16)(2,9,12,28)(4,19,20,29)(5,15,6,14)(8,22,24,32)(10,27,11,26)(13,17,30,21)(18,25,23,31)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=egh,g^c=gi,h^c=hi,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72231251, C2×#143163, #72306765, … |
| maximal normal subgroup of | #72231251, C2×#143163, #72306765, … |
| maximal quotient of | C2×#143163, #72309248 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50359);