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 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(2,6)(3,8)(11,19)(13,23)(15,24)(17,20)(25,26,31,28)(27,32,30,34)(29,33,38,36)(35,39,37,40);(2,6)(4,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,28,31,26)(27,34,30,32)(29,33,38,36)(35,39,37,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 10235576);