Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489351 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18524 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
903 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 903 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:
(2,7)(3,11)(4,14)(5,17)(8,22)(9,26)(10,15)(12,29)(13,30)(16,19)(20,27)(21,24)(23,28)(25,32);(1,2,6,10)(3,8,12,23)(4,15,14,7)(5,18,20,31)(9,27,26,17)(11,28,29,22)(13,21,16,25)(19,24,30,32);(1,3)(2,8)(4,16)(5,19)(6,12)(7,21)(9,28)(10,23)(11,17)(13,14)(15,25)(18,24)(20,30)(22,26)(27,29)(31,32);(5,20)(9,26)(11,29)(17,27)(18,31)(19,30)(22,28)(24,32)Polycyclic presentation:
2^9|b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72214208, #72215848, #72215869, … |
| maximal normal subgroup of | #72214208, #72215848, #72215869, … |
| maximal quotient of | #72224810, #72226787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396548);