Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(17,18)(19,25)(20,22)(21,24)(23,27)(26,30)(28,32)(29,31);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,21)(18,24,27,25)(20,28,30,31)(22,29,26,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,19,23,21)(18,25,27,24)(20,28,30,31)(22,32,26,29);(11,15)(13,16)(17,20)(18,26)(19,28)(21,31)(22,27)(23,30)(24,32)(25,29);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,19)(18,24,27,25)(20,28,30,31)(22,32,26,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22,23,26)(18,20,27,30)(19,29,21,32)(24,28,25,31)Polycyclic presentation:
2^9|b^2=g,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72265951, #72265956, #72265962, … |
| maximal normal subgroup of | #72265951, #72265956, #72265962, … |
| maximal quotient of | #72266508, #72266556, #72300342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805278);