Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485052 |
| 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×C4⋊C4 | C24 |
| prime radical | #23948 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1029 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)(3,15)(4,21)(5,25)(6,11)(7,27)(8,14)(9,17)(10,22)(12,16)(13,20)(18,32)(19,28)(23,31)(24,29)(26,30);(1,3,7,19)(2,9,27,21)(4,16,17,26)(5,24,22,23)(6,18,8,20)(10,28,25,15)(11,29,14,31)(12,32,30,13);(1,4)(2,10)(3,16)(5,18)(6,23)(7,17)(8,24)(9,28)(11,30)(12,14)(13,29)(15,21)(19,26)(20,22)(25,27)(31,32);(2,11)(9,29)(10,30)(12,25)(13,28)(14,27)(15,32)(21,31)Polycyclic presentation:
2^9|b^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,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 | #72240300, #72278749, #72338703, … |
| maximal normal subgroup of | #72240300, #72278749, #72338703, … |
| maximal quotient of | #72240577 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392287);