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 | |
4581 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)(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,21)(22,24)(25,26)(27,29)(28,33)(30,34)(31,35)(32,38)(36,37)(39,40);(3,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,17)(18,22)(23,24)(25,27,31,38)(26,32,35,29)(28,36,34,40)(30,37,33,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(13,23)(20,22)(28,34)(30,33)(36,40)(37,39);(9,11)(10,17)(12,22)(13,21)(14,19)(15,16)(18,20)(23,24)(25,28,31,34)(26,33,35,30)(27,36,38,40)(29,37,32,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,18)(10,13,16,23)(11,20,19,22)(15,21,17,24)(25,29,31,32)(26,27,35,38)(28,39,34,37)(30,36,33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,21)(12,16)(14,23)(15,22)(17,20)(19,24)(25,30,31,33)(26,34,35,28)(27,37,38,39)(29,36,32,40)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837865);