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 | |
5799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,15,16,12)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(1,2)(3,7)(4,5)(6,8)(10,14)(11,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(2,5)(6,7)(10,14)(12,15)(21,29)(24,31)(26,32)(28,30);(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^9|a^2=g,b^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72327643 |
| maximal normal subgroup of | #72327643 |
| maximal quotient of | #72327815, #72328539, #72329700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8106836);