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 | |
4434 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)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,19)(14,15)(16,18)(17,20)(21,24)(22,23)(27,36)(29,37)(32,35)(34,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,15,18,11)(13,21,22,20)(17,19,24,23)(25,27,31,36)(26,32,30,35)(28,29,39,37)(33,40,38,34);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,22)(23,24)(25,28,31,39)(26,33,30,38)(27,37,36,29)(32,34,35,40);(9,11)(10,16)(12,18)(13,19)(14,15)(17,21)(20,24)(22,23)(25,29,31,37)(26,34,30,40)(27,28,36,39)(32,38,35,33);(9,13,14,22)(10,17,18,24)(11,19,15,23)(12,20,16,21)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8937072);