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 | |
4337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,20,17)(13,18,22,16)(19,24,21,23)(26,30)(28,36)(31,33)(34,40);(2,3)(6,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(23,24)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,5,2)(4,6,8,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755582);