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 | |
4301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,4)(6,7)(10,15)(13,23)(16,21)(17,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229447);