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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(11,17)(12,20)(15,18)(16,21)(25,26,30,35)(27,34,29,31)(28,33,38,32)(36,39,37,40);(3,7)(6,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,30,29)(26,31,35,34)(28,36,38,37)(32,39,33,40);(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(16,21)(19,22)(25,28,30,38)(26,32,35,33)(27,36,29,37)(31,39,34,40);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28)(26,33)(27,37)(29,36)(30,38)(31,39)(32,35)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,30,27)(26,34,35,31)(28,37,38,36)(32,40,33,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840406);