Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,26,28,29)(27,31,32,30);(10,12)(11,16)(17,18,21,22)(19,20,24,23)(26,29)(30,31);(10,12)(14,15)(18,22)(20,23)(25,26,28,29)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,19,22,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72326688, #72326709, #72326728, … |
| maximal normal subgroup of | #72326688, #72326709, #72326728, … |
| maximal quotient of | #72328988, #72329879, #72330499 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5082513);