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 | |
930 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 930 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)(3,11)(4,16)(5,21)(6,8)(7,13)(9,17)(10,20)(12,26)(14,28)(15,27)(18,23)(19,31)(22,30)(24,29)(25,32);(4,17)(5,20)(9,16)(10,21)(11,26)(12,15)(13,18)(19,32)(23,24)(25,31)(27,28)(29,30);(1,3)(2,7)(4,18)(5,12)(6,14)(8,22)(9,26)(10,23)(11,16)(13,17)(15,20)(19,29)(21,24)(25,27)(28,31)(30,32);(2,8)(4,19)(7,22)(10,21)(13,30)(17,32)(18,29)(23,24)Polycyclic presentation:
2^9|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, #72215809, #72215848, … |
| maximal normal subgroup of | #72214208, #72215809, #72215848, … |
| maximal quotient of | #72219637, #72224812, #72226787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396547);