Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8,10,22)(3,13)(4,17)(5,18)(9,24,26,31)(11,23,12,19)(14,28)(15,30)(16,29)(21,25,32,27);(1,2)(3,9)(4,8)(5,19)(6,11)(7,12)(10,20)(13,27)(14,25)(15,21)(16,24)(17,23)(18,22)(26,28)(29,31)(30,32);(1,3)(2,9)(4,15)(5,16)(6,14)(7,13)(8,21)(10,26)(11,25)(12,27)(17,29)(18,30)(19,24)(20,28)(22,32)(23,31);(3,14)(9,25)(13,28)(15,29)(16,30)(21,31)(24,32)(26,27);(1,4,7,5)(2,8,12,19)(3,15,13,16)(6,17,20,18)(9,21,27,24)(10,22,11,23)(14,29,28,30)(25,31,26,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=bf,c^a=ci,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232917, #72233035, #72233128, … |
| maximal normal subgroup of | #72232917, #72233035, #72233128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250597);