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 | |
4463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23);(3,7)(6,8)(10,15)(11,18)(12,21)(23,24)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,2,5,4)(3,6,7,8)(10,15)(13,16)(17,20)(23,24)(26,31)(29,33)(32,39)(35,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,40)(34,38);(1,4,5,2)(3,8,7,6)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,29,30,33)(26,28,31,38)(27,35,36,40)(32,37,39,34)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279401);