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 | |
5213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,13,14)(11,12,15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31);(3,4)(6,8)(10,14)(11,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,15)(14,16)(18,25)(19,20)(26,32)(27,28);(3,4)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31);(1,4,5,3)(2,6,7,8)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32);(1,2)(3,6)(4,8)(5,7)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10090263);