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 | |
4922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,6,8,7)(10,13)(14,16)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32);(1,2,5,4)(3,7,8,6)(18,20)(24,25)(26,28)(31,32);(1,3)(2,7)(4,6)(5,8)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,4,5,2)(3,6,8,7)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=hi,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, 7893135);