Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3685 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)(10,11)(14,15)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,31)(29,32);(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,23)(22,24)(25,27)(26,29)(28,32)(30,31);(3,7)(6,8)(10,11)(14,15)(18,19)(22,23)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=fghi,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 | #72232632, #72232653, #72232659, … |
| maximal normal subgroup of | #72232632, #72232653, #72232659, … |
| maximal quotient of | #72328509, #72328663, #72328724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1490605);