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 | |
5422 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)(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,2,5,8)(3,6,7,4)(18,22)(23,25)(24,28)(30,31);(1,4)(2,7)(3,8)(5,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4)(2,3)(5,6)(7,8)(9,12,13,14)(10,11,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72330757 |
| maximal normal subgroup of | #72330757 |
| maximal quotient of | #72330826, #72331803, #72331981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592793);