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 |
| prime radical | #44457 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(23,24)(25,26)(27,30)(28,29)(31,32)(33,39)(34,35)(36,38)(37,40);(2,3)(6,7)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,27,31,33)(26,30,32,39)(28,34,36,37)(29,35,38,40);(1,2,5,3)(4,6,8,7)(26,32)(27,33)(28,36)(35,40);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28)(26,29)(27,34)(30,35)(31,36)(32,38)(33,37)(39,40);(1,3,5,2)(4,6,8,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29)(26,28)(27,35)(30,34)(31,38)(32,36)(33,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30)(26,33)(27,32)(28,35)(29,37)(31,39)(34,38)(36,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9755884);