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 | |
4868 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)(9,10,14,15)(11,16,12,19)(13,17,21,22)(18,23,20,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,3)(6,7)(10,15)(16,19)(17,22)(23,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11,14,12)(10,16,15,19)(13,18,21,20)(17,23,22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,4,5,8)(2,6,3,7)(9,12,14,11)(10,16,15,19)(13,18,21,20)(17,24,22,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,10)(11,16)(12,19)(13,17)(14,15)(18,23)(20,24)(21,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315328);