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 | |
4615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,3,5,8)(2,6,7,4)(11,17)(15,18)(19,23)(22,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3)(2,4)(5,8)(6,7)(10,12)(15,18)(16,20)(22,24)(29,37)(33,40)(35,36)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002971);