Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2974 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,22,23,20)(25,27)(26,31)(28,29)(30,32);(3,7)(6,8)(11,15)(14,16)(18,23)(20,22)(26,29)(27,32);(1,2)(3,8)(4,5)(6,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,22)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|b^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72328079, #72328094, #72328235, … |
| maximal normal subgroup of | #72328079, #72328094, #72328235, … |
| maximal quotient of | #72331386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3967005);