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 | |
4882 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)(3,8)(4,6)(5,7)(10,12)(13,15)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32);(2,4)(7,8)(10,12)(11,16)(18,20)(19,25)(24,28)(27,31);(1,3,6,5)(2,7,4,8)(9,10,14,12)(11,15,16,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31);(1,2,6,4)(3,7,5,8)(9,11,14,16)(10,15,12,13)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29);(1,4,6,2)(3,8,5,7)(9,12,14,10)(11,13,16,15)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32);(1,5,6,3)(2,8,4,7)(9,13)(10,11)(12,16)(14,15)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015944);