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 | |
5346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,23,25)(19,26,21,24)(20,29,31,22)(27,28,30,32);(1,2)(3,6)(4,7)(5,8)(10,13)(11,15)(17,19,23,21)(18,24,25,26)(20,27,31,30)(22,28,29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,29)(26,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,19)(18,24,25,26)(20,30,31,27)(22,28,29,32);(1,2)(3,5)(4,7)(6,8)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,31)(26,30)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72317548, #72317591, #72317637, … |
| maximal normal subgroup of | #72317548, #72317591, #72317637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003022);