Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484583 |
| 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 | #36725 | 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 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,13,16,27)(4,10,5,18)(7,12,21,25)(8,19,24,14)(9,20,26,17)(15,28,30,22)(23,31,32,29);(1,3)(2,8)(4,14)(5,19)(6,16)(7,15)(9,22)(10,27)(11,24)(12,23)(13,18)(17,29)(20,31)(21,30)(25,32)(26,28);(1,4,6,5)(2,9,25,18)(3,14,16,19)(7,17,21,20)(8,22,32,13)(10,11,26,12)(15,29,30,31)(23,27,24,28);(3,15)(8,23)(13,28)(14,29)(16,30)(19,31)(22,27)(24,32)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,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 | #72285299, #72285778, #72285796, … |
| maximal normal subgroup of | #72285299, #72285778, #72285796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392024);