Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485315 |
| 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 |
| prime radical | #18162 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,12)(4,7)(5,9)(6,11)(8,13)(10,16)(14,22)(15,27)(17,20)(18,21)(19,23)(24,29)(25,30)(26,31)(28,32);(2,7)(3,13)(4,16)(8,22)(9,10)(11,21)(12,20)(14,30)(15,29)(17,32)(18,31)(23,26)(24,28)(25,27);(1,3,5,15)(2,8,23,27)(4,17,16,30)(6,13,19,29)(7,20,26,28)(9,24,11,12)(10,25,21,22)(14,31,32,18);(2,9)(7,10)(8,24)(11,23)(12,27)(20,25)(21,26)(22,28)Polycyclic presentation:
2^9|c^2=ghi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251770, #72252026, #72252027 |
| maximal normal subgroup of | #72251770, #72252026, #72252027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392586);