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 | |
5100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,6,7)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(20,28)(30,31);(1,3)(2,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,26,24)(20,25,28,32)(27,31,29,30);(1,4)(2,6)(3,8)(5,7)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,31,32,30);(1,3)(2,7)(4,8)(5,6)(17,19)(18,21)(20,27)(22,26)(23,24)(25,31)(28,29)(30,32);(1,3)(2,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,27)(21,31)(22,28)(23,32)(24,30)(26,29);(1,4)(2,6)(3,8)(5,7)(17,21)(18,19)(20,30)(22,24)(23,26)(25,29)(27,32)(28,31)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875597);