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 | |
4665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(3,7)(6,8)(10,16)(11,12)(18,23)(19,20)(26,28)(29,34)(33,36)(37,40);(1,2,5,4)(3,6,7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,3,5,7)(2,6,4,8)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(1,4,5,2)(3,8,7,6)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400437);