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 | |
4850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(2,3)(6,7)(10,14)(15,16)(18,19)(23,25)(24,26)(30,31);(1,2,5,3)(4,6,8,7)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(9,11,13,12)(10,15,14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017190);