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 | |
4957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,12)(14,15)(17,18,22,24)(19,25,20,23)(21,26,30,29)(27,32,28,31);(1,3,5,8)(2,6,4,7)(17,19,22,20)(18,23,24,25)(21,28,30,27)(26,32,29,31);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,15,16,14)(18,24)(19,20)(26,29)(27,28);(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31);(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31);(1,4,5,2)(3,7,8,6)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=gi,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, 7895414);