Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486097 |
| 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 | |
1025 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1025 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)(3,16)(4,12)(5,8)(6,13)(7,15)(9,19)(10,22)(11,18)(14,26)(17,32)(20,29)(21,31)(23,30)(24,27)(25,28);(2,8)(3,17)(4,21)(5,18)(9,14)(10,27)(11,13)(12,32)(15,28)(16,30)(19,25)(20,24)(22,23)(29,31);(1,3,7,17)(2,9,15,11)(4,19,23,18)(5,22,25,21)(6,20,26,24)(8,13,28,14)(10,29,31,27)(12,16,30,32);(1,4)(2,10)(3,18)(5,20)(6,22)(7,23)(8,16)(9,27)(11,29)(12,13)(14,30)(15,31)(17,19)(21,26)(24,25)(28,32)Polycyclic presentation:
2^9|c^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72232556, #72234738, #72237867 |
| maximal normal subgroup of | #72232556, #72234738, #72237867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393307);