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 | #36785 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(2,3)(6,7)(10,15)(11,18)(16,21)(19,24)(25,27,31,38)(26,32,35,34)(28,36,39,29)(30,37,40,33);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,23)(22,24)(25,28,31,39)(26,33,35,37)(27,36,38,29)(30,34,40,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,30)(27,28)(31,36)(32,37)(33,34)(35,40)(38,39);(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30)(26,29)(27,37)(28,32)(31,40)(33,38)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833840);