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 | |
4593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,2)(3,4)(5,7)(6,8)(10,14)(11,16)(19,25)(22,29)(26,31)(30,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=hi,d^2=hi,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, 8658025);