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)(3,4)(5,8)(6,7)(9,10,13,15)(11,12,16,14)(20,28)(23,30)(27,32)(29,31);(1,3)(2,4)(5,6)(7,8)(9,11,13,16)(10,14,15,12)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32);(1,2,7,6)(3,5,8,4)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,29)(24,26)(28,32)(30,31);(1,4,7,5)(2,8,6,3)(9,12)(10,11)(13,14)(15,16)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,32)(26,30);(1,5,7,4)(2,8,6,3)(17,19)(18,24)(20,27)(21,26)(22,25)(23,31)(28,32)(29,30);(1,6,7,2)(3,5,8,4)(17,21,22,24)(18,25,26,19)(20,29,28,31)(23,32,30,27)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619997);