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 | |
4939 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)(3,4,7,6)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,22)(21,28)(24,29)(25,26)(27,32)(30,31);(1,2)(3,6)(4,7)(5,8)(10,14)(11,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32);(3,7)(4,6)(9,11)(10,12)(13,15)(14,16)(18,20)(19,25)(24,28)(27,31);(1,3,5,7)(2,6,8,4)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,5,6)(2,7,8,3)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,2)(3,6)(4,7)(5,8)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038423);