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 | |
5470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,21,29,24)(20,30,28,25)(22,26,32,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,21)(20,22,28,32)(25,26,30,31);(2,3)(6,7)(10,12)(11,15)(17,20,23,28)(18,25,27,30)(19,22,29,32)(21,31,24,26);(4,8)(6,7)(11,15)(13,16)(17,19)(18,24)(20,22)(21,27)(23,29)(25,26)(28,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,15,16)(17,21,23,24)(18,19,27,29)(20,26,28,31)(22,25,32,30);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,22)(18,26)(19,28)(20,29)(21,25)(23,32)(24,30)(27,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519104);