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 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,26)(23,25)(27,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(23,25)(24,28)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249479, #72249486, #72249540, … |
| maximal normal subgroup of | #72249479, #72249486, #72249540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550760);