Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
5084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,23,28,32)(20,24,30,21)(26,27,29,31);(1,3,6,4)(2,7,5,8)(10,13)(14,15)(17,19,22,28)(18,23,25,32)(20,29,30,26)(21,27,24,31);(2,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,31)(22,28)(24,27)(25,32)(29,30);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,26)(21,25)(22,30)(23,27)(28,29)(31,32);(1,4,6,3)(2,8,5,7);(1,5,6,2)(3,8,4,7)(17,21,22,24)(18,20,25,30)(19,27,28,31)(23,29,32,26)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311332, #72311343, #72311396, … |
| maximal normal subgroup of | #72311332, #72311343, #72311396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103218);