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 |
| prime radical | #39282 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,29)(27,31,35,33)(28,36,38,32)(34,40,37,39);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,29)(27,33,35,31)(28,32,38,36)(34,40,37,39);(1,2,5,4)(3,6,7,8)(12,20)(16,21)(18,24)(22,23)(26,29)(31,33)(32,36)(39,40);(1,3,5,7)(2,6,4,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,37)(29,33)(30,35)(32,40)(34,38)(36,39);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,38)(26,32,29,36)(27,34,35,37)(31,39,33,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,26)(27,33,35,31)(28,36,38,32)(34,40,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=h,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, 8247045);