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 | #39548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4438 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)(3,4,8,6)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(27,28)(31,34)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(13,21)(17,24)(19,20)(22,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8028662);