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 | |
4543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,17)(13,22)(14,15)(16,19)(20,21)(23,24)(25,26,30,32)(27,33,35,29)(28,31,36,39)(34,40,37,38);(1,2)(3,7)(4,5)(6,8)(10,15)(11,19)(13,17)(20,24)(25,27,30,35)(26,29,32,33)(28,34,36,37)(31,38,39,40);(2,5)(6,7)(10,15)(12,22)(16,18)(20,24)(28,36)(31,39)(34,37)(38,40);(2,5)(6,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(25,27)(26,29)(28,37)(30,35)(31,40)(32,33)(34,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,28,30,36)(26,31,32,39)(27,34,35,37)(29,38,33,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)(25,29)(26,27)(28,38)(30,33)(31,34)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,e^2=gh,f^2=h,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, 10230870);