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 | |
5431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30);(3,7)(5,8)(10,11)(14,15)(17,19,21,27)(18,22,23,25)(20,26,29,28)(24,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(18,23)(22,25)(24,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31);(1,2)(3,5)(4,6)(7,8)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72330757 |
| maximal normal subgroup of | #72330757 |
| maximal quotient of | #72330855, #72331835, #72332005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8438839);