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 | |
5066 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)(3,7)(4,8)(5,6)(10,12)(13,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,32)(28,31);(2,6)(4,7)(10,12)(11,16)(18,23)(20,25)(24,26)(27,31);(1,3)(2,4)(5,8)(6,7)(9,10,14,12)(11,15,16,13);(1,4)(2,3)(5,7)(6,8)(9,11,14,16)(10,15,12,13)(17,19,22,29)(18,24,23,26)(20,27,25,31)(21,28,30,32);(1,4)(2,3)(5,7)(6,8)(9,12,14,10)(11,13,16,15)(17,20,22,25)(18,21,23,30)(19,27,29,31)(24,28,26,32);(1,3)(2,4)(5,8)(6,7)(9,13)(10,11)(12,16)(14,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,31)(26,27)(29,32)Polycyclic presentation:
2^9|c^2=i,d^2=gh,e^2=gh,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, 8015940);