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 | |
4958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,40)(35,38);(2,6)(3,7)(10,14)(11,18)(16,22)(19,21)(25,27,30,33)(26,28,31,36)(29,38,39,34)(32,40,37,35);(1,3)(2,4)(5,7)(6,8)(12,20)(16,22)(19,21)(23,24)(26,31)(27,33)(32,37)(34,38);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,32)(27,34)(28,35)(30,39)(31,37)(33,38)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^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, 10231795);