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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26,30,33)(27,31,35,28)(29,37,39,32)(34,36,40,38);(1,3)(2,5)(4,8)(6,7)(10,14)(11,17)(16,19)(18,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,4,6,5)(2,7,8,3)(25,27,30,35)(26,28,33,31)(29,34,39,40)(32,36,37,38);(1,2)(3,4)(5,7)(6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(1,2)(3,5)(4,7)(6,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,39)(26,32,33,37)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852751);