Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2955 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26,29,28)(27,31,32,30);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20)(25,27)(26,30)(28,31)(29,32);(3,8)(6,7)(11,15)(13,16)(18,23)(20,22)(25,27,29,32)(26,31,28,30);(1,3,4,8)(2,6,5,7)(9,11)(10,13)(12,15)(14,16)(17,20,21,22)(18,19,23,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,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 | #72266509, #72266512, #72266520, … |
| maximal normal subgroup of | #72266509, #72266512, #72266520, … |
| maximal quotient of | #72334834 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1623434);