Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470368 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,26,29,32)(21,25,22,31)(24,28,27,30);(9,11)(10,13)(12,15)(14,16)(17,19,22,30,23,29,21,28)(18,24,25,26,20,27,31,32);(2,5)(6,8)(11,15)(13,16)(17,20,23,18)(19,27,29,24)(21,25,22,31)(26,30,32,28);(11,15)(13,16)(17,21,23,22)(18,25,20,31)(19,28,29,30)(24,26,27,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,29)(24,27)(26,32)(28,30)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284689, #72284732, #72284961, … |
| maximal normal subgroup of | #72284689, #72284732, #72284961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383742);