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 | #59212 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(10,15)(11,13)(12,20)(23,24)(25,26)(27,32)(28,37)(29,39)(30,31)(33,38)(34,35)(36,40);(2,3)(6,7)(9,10,14,15)(11,16,13,18)(12,21,20,17)(19,24,22,23)(26,31)(32,40)(33,37)(34,39);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,15,18)(12,19,20,22)(17,23,21,24)(25,27,30,36)(26,32,31,40)(28,35,38,29)(33,39,37,34);(9,12,14,20)(10,17,15,21)(11,19,13,22)(16,23,18,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,3,5,2)(4,7,8,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,18,15,16)(12,22,20,19)(17,24,21,23)(25,29,30,35)(26,34,31,39)(27,28,36,38)(32,33,40,37)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017160);