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 | |
4866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,15)(11,12,16,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(21,29)(24,31)(26,32)(28,30);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32);(1,4,5,2)(3,8,7,6)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(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=g,c^2=h,d^2=ghi,e^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, 8309514);