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 | |
4316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,20,21,15)(12,17,22,16)(13,23,24,18)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,40)(36,38);(1,2,5,6)(3,4,7,8)(9,11,14,21)(10,15,19,20)(12,18,22,23)(13,17,24,16)(25,27,30,35)(26,28,32,33)(29,38,39,34)(31,40,37,36);(2,6)(3,7)(9,12,14,22)(10,16,19,17)(11,18,21,23)(13,20,24,15)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,40,38,36);(2,6)(3,7)(9,11)(10,15)(12,18)(13,16)(14,21)(17,24)(19,20)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,20)(14,22)(15,24)(16,19)(21,23)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,19,23)(11,16,21,17)(12,20,22,15)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,f^2=g,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, 10235732);