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 | |
4827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,13)(14,16)(19,25)(23,30)(26,28)(31,32);(2,6)(4,7)(10,13)(14,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,27)(26,28);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31);(1,4)(2,3)(5,7)(6,8)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32);(1,3)(2,4)(5,8)(6,7)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=gi,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, 8838208);