Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485315 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #23984 | 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 | |
1298 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1298 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)(3,13)(4,11)(5,18)(6,10)(7,12)(8,14)(9,17)(15,28)(16,27)(19,23)(20,26)(21,25)(22,30)(24,29)(31,32);(1,3,7,15)(2,8,12,24)(4,14,17,29)(5,19,20,31)(6,16,21,30)(9,22,11,27)(10,23,25,32)(13,26,28,18);(1,4,6,5)(2,9,25,18)(3,14,16,19)(7,17,21,20)(8,22,32,13)(10,26,12,11)(15,29,30,31)(23,28,24,27);(2,10)(8,23)(9,26)(11,18)(12,25)(13,27)(22,28)(24,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,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 | #72227860, #72231830, #72240204, … |
| maximal normal subgroup of | #72227860, #72231830, #72240204, … |
| maximal quotient of | #72227852, #72240381, #72278468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392580);