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 | |
5130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32);(1,2,5,4)(3,7,8,6)(10,13)(14,15)(20,27)(24,30)(26,31)(28,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,25)(24,30)(26,31);(1,2,5,4)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32);(1,4,5,2)(3,7,8,6)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32);(1,3)(2,7)(4,6)(5,8)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9773091);