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 | |
4559 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)(3,4)(5,8)(6,7)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,23,21,20)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(2,7)(4,5)(9,11)(10,15)(12,17)(13,21)(14,19)(16,20)(18,22)(23,24)(25,26,30,29)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,3,6,8)(2,5,7,4)(9,11,14,19)(10,15,17,12)(13,18,22,21)(16,23,24,20)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39);(1,4,6,5)(2,8,7,3)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,29,30,26)(27,33,35,31)(28,32,36,37)(34,38,40,39);(1,5,6,4)(2,3,7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=g,e^2=gh,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, 10247432);