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 | #42602 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4305 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(2,6)(3,8)(9,11,15,12)(10,16,19,22)(13,20,18,23)(14,21,17,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3)(2,4)(5,8)(6,7)(11,12)(16,22)(20,23)(21,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,16,19,22)(13,23,18,20)(14,21,17,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,20)(12,23)(14,19)(15,18)(16,24)(21,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(9,14,15,17)(10,18,19,13)(11,21,12,24)(16,23,22,20)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8658073);