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 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(12,19)(16,21)(20,23)(26,31)(29,32)(33,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,18,19)(13,21,22,16)(17,23,24,20);(2,5)(6,7)(9,11,14,18)(10,12,15,19)(13,17,22,24)(16,20,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(2,5)(6,7)(9,12,14,19)(10,11,15,18)(13,20,22,23)(16,17,21,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)(25,28,30,38)(26,33,31,39)(27,35,34,36)(29,37,32,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,16,15,21)(11,17,18,24)(12,20,19,23)(25,29,30,32)(26,34,31,27)(28,37,38,40)(33,36,39,35)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231212);