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 | #36675 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,2)(3,4)(5,7)(6,8)(10,11)(15,17)(16,18)(22,23)(26,27)(28,35)(32,34)(36,40);(1,2)(3,4)(5,7)(6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3,5,8)(2,6,7,4)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658484);