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 |
| prime radical | #49564 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26)(27,33)(28,36)(29,35)(30,31)(32,39)(34,38)(37,40);(2,3)(6,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27)(26,29)(28,37)(30,35)(31,33)(32,40)(34,39)(36,38);(1,3,5,2)(4,7,8,6)(10,14)(12,21)(15,17)(18,22)(26,31)(29,33)(32,36)(38,40);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,30,39)(26,32,31,36)(27,34,35,37)(29,38,33,40);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,26)(27,29)(28,36)(30,31)(32,39)(33,35)(34,40)(37,38);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947488);