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 | #42480 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,5)(3,6,8,7)(10,14)(15,16)(18,23)(19,26)(25,29)(27,32);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,12,16)(17,18,22,23)(19,24,26,20)(21,29,31,25)(27,28,32,30);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(18,23)(20,24)(25,29)(28,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,30)(28,29)(31,32);(1,2)(3,7)(4,5)(6,8)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,32,29,27);(1,3,4,8)(2,6,5,7)(17,21,22,31)(18,25,23,29)(19,27,26,32)(20,28,24,30)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=gi,e^2=hi,f^2=i,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, 7896818);