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 | |
4520 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(2,5)(6,7)(10,15)(12,22)(16,18)(19,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,23)(13,24)(14,21)(15,18)(17,20)(19,22)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,12)(10,17)(11,19)(13,15)(14,22)(16,24)(18,20)(21,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,19,18,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353693);