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 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,2,5,4)(3,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,27)(26,31)(28,32)(29,35)(30,33)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(10,14)(15,21)(16,19)(22,24)(26,32)(27,33)(29,38)(36,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,28)(29,35)(30,33)(31,32)(34,36)(37,39)(38,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,33)(27,32)(29,36)(30,31)(34,40)(35,37)(38,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385362);