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 | #42532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,16,15)(11,13,17,12)(14,22,23,18)(19,20,24,21)(28,36)(31,39)(35,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,19)(16,17)(18,20)(21,22)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(9,12,16,13)(10,17,15,11)(14,20,23,21)(18,24,22,19)(25,27)(26,29)(28,35)(30,33)(31,38)(32,34)(36,40)(37,39);(1,3)(2,7)(4,6)(5,8)(9,13,16,12)(10,17,15,11)(14,21,23,20)(18,24,22,19)(25,28,30,36)(26,31,34,39)(27,35,33,40)(29,38,32,37);(1,3)(2,7)(4,6)(5,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,27)(26,32)(28,35)(29,34)(30,33)(31,37)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,15,16,10)(11,13,17,12)(14,18,23,22)(19,20,24,21)(25,29,30,32)(26,33,34,27)(28,37,36,38)(31,35,39,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595523);