Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26,28,30)(27,31,32,29);(2,6)(4,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,20,22,23);(2,6)(3,7)(11,16)(13,14)(17,18,21,22)(19,23,24,20)(25,27,28,32)(26,29,30,31);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72334248, #72334252, #72334254, … |
| maximal normal subgroup of | #72334248, #72334252, #72334254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1038732);