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 | |
4640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,27)(20,28)(22,30)(24,32);(9,11,12,16)(10,14,13,15)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31);(1,3)(2,5)(4,7)(6,8)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32);(1,2)(3,5)(4,6)(7,8)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835842);