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 | |
4574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,8)(9,10,14,15)(11,13,12,16)(19,25)(20,27)(22,28)(23,30);(1,2)(3,7)(4,8)(5,6)(10,15)(11,12)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,16,15,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(2,6)(4,7)(9,12,14,11)(10,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658491);