Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,19)(18,24,26,27)(21,28,31,29)(23,30,32,25);(3,10)(7,14)(11,16)(13,15)(19,20)(23,32)(24,27)(28,29)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72292039 |
| maximal normal subgroup of | #72292039 |
| maximal quotient of | #72292050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394002);