Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486077 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18178 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 763 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,8)(4,7)(5,19)(6,11)(9,15)(10,18)(12,27)(13,30)(14,25)(16,21)(17,20)(22,31)(23,28)(24,29)(26,32);(2,7)(3,12)(4,15)(5,18)(8,23)(9,10)(11,21)(13,17)(14,29)(16,31)(19,22)(20,24)(25,26)(27,30);(4,16)(13,17)(15,31)(28,32);(1,3,5,12,6,14,18,29)(2,8,19,27,11,25,10,24)(4,17,15,32,16,13,31,28)(7,20,9,26,21,30,22,23)Polycyclic presentation:
2^9|d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,e^d=ei,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231710, C2×#486080, #72281886, … |
| maximal normal subgroup of | #72231710, C2×#486080, #72281886, … |
| maximal quotient of | #72250365, C2×#486080, #72279792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393276);