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 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,17)(11,13,16,18)(12,15,23,21)(19,22,20,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(2,3)(6,7)(9,11)(10,13)(12,20)(14,16)(15,24)(17,18)(19,23)(21,22)(26,28)(27,34)(32,36)(35,40);(1,3,5,2)(4,7,8,6)(11,16)(13,18)(19,20)(22,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,21)(18,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,13,14,18)(10,16,17,11)(12,22,23,24)(15,20,21,19)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325594);