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 | |
4576 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)(3,4)(5,8)(6,7)(9,10,14,15)(11,16,13,12)(17,18,22,19)(20,23,27,25)(21,24,30,26)(28,31,29,32);(1,3,7,8)(2,4,6,5)(9,11,14,13)(10,12,15,16)(18,19)(20,27)(24,26)(28,29);(1,2,7,6)(3,5,8,4)(9,11,14,13)(10,12,15,16);(1,4,7,5)(2,8,6,3)(9,12)(10,11)(13,15)(14,16)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31);(1,5,7,4)(2,8,6,3)(9,13,14,11)(10,12,15,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30);(1,6,7,2)(3,5,8,4)(9,12)(10,13)(11,15)(14,16)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9842982);