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 | |
4574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,6)(3,7,8,4)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(2,6)(4,7)(10,14)(16,18)(17,20)(23,24)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(9,13,15,11)(10,18,14,16)(12,21,22,19)(17,24,20,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4)(2,3)(5,7)(6,8)(9,14,15,10)(11,18,13,16)(12,20,22,17)(19,24,21,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8088352);