Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | #14722 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26)(27,31)(28,29)(30,32);(2,5)(3,8)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20)(26,29)(27,32);(2,5)(6,7)(11,15)(13,16)(18,23)(20,22)(25,26,28,29)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,20,21,22)(18,19,23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303423, #72303506, #72303512, … |
| maximal normal subgroup of | #72303423, #72303506, #72303512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1723261);