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 | #35539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(11,18)(12,15)(19,23)(20,21)(25,26,32,36)(27,33,38,28)(29,40,37,34)(30,35,39,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,27)(26,33)(28,36)(29,35)(30,34)(31,37)(32,38)(39,40);(9,11,14,18)(10,15,17,12)(13,19,22,23)(16,21,24,20)(25,28)(26,27)(29,39)(30,37)(31,40)(32,33)(34,35)(36,38);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,29,32,37)(26,34,36,40)(27,35,38,31)(28,39,33,30);(9,12)(10,11)(13,21)(14,15)(16,23)(17,18)(19,24)(20,22)(25,30,32,39)(26,35,36,31)(27,34,38,40)(28,37,33,29);(9,13,14,22)(10,16,17,24)(11,19,18,23)(12,20,15,21)(25,31,32,35)(26,30,36,39)(27,37,38,29)(28,40,33,34)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825866);