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 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4796 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)(10,13)(14,16)(17,18,23,24)(19,20,28,25)(21,30,32,26)(22,27,31,29);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,28)(22,31)(26,30);(1,4)(2,6)(3,8)(5,7)(17,19,23,28)(18,25,24,20)(21,31,32,22)(26,29,30,27);(1,3)(2,7)(4,8)(5,6)(17,20,23,25)(18,19,24,28)(21,29,32,27)(22,30,31,26);(1,3)(2,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,23,32)(18,26,24,30)(19,22,28,31)(20,29,25,27);(1,4)(2,6)(3,8)(5,7)(17,22)(18,27)(19,21)(20,30)(23,31)(24,29)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=g,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, 7875257);