Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485637 |
| 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 | #60045 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 980 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:
(1,2,7,13)(3,15,17,27)(4,12,20,31)(5,10,16,11)(6,24,21,32)(8,25,9,14)(18,26,19,29)(22,30,23,28);(2,10)(4,20)(5,16)(6,21)(11,13)(12,29)(14,28)(15,24)(22,23)(25,30)(26,31)(27,32);(1,3)(2,11)(4,16)(5,20)(6,22)(7,17)(8,18)(9,19)(10,13)(12,29)(14,30)(15,24)(21,23)(25,28)(26,31)(27,32);(1,4,9,6)(2,12,14,24)(3,16,19,22)(5,18,23,17)(7,20,8,21)(10,26,28,27)(11,29,30,15)(13,31,25,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209805, #72209807, #72278749, … |
| maximal normal subgroup of | #72209805, #72278749, #72278795, … |
| maximal quotient of | #72227643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392916);