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 | |
6460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,14,12)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30);(9,11)(10,12)(13,14)(15,16)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32);(2,5)(6,7)(9,12,13,16)(10,14,15,11)(18,23)(19,26)(21,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,14)(15,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72235981, #72251351, #72265947, … |
| maximal normal subgroup of | #72235981, #72251351, #72265947, … |
| maximal quotient of | #72235983, #72235988, #72333437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713797);