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 | |
4898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(18,23)(20,27)(25,29)(28,32);(1,2)(3,4)(5,6)(7,8)(10,13)(14,15)(18,23)(20,27)(25,29)(28,32);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(2,6)(3,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,32)(30,31);(1,3)(2,7)(4,6)(5,8)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4,5,7)(2,3,6,8)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^9|c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8305762);