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 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26,31,28)(27,32,34,29)(30,36,38,33)(35,37,40,39);(3,7)(6,8)(10,15)(11,18)(16,21)(19,24)(25,27,31,34)(26,32,28,29)(30,35,38,40)(33,39,36,37);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,21)(20,23)(27,34)(29,32)(35,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,23)(22,24)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30,31,38)(26,33,28,36)(27,35,34,40)(29,37,32,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032629);