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 | #59020 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,17)(12,16)(13,18)(14,15)(19,21)(20,22)(23,24)(25,26,30,34)(27,31,37,40)(28,32,36,39)(29,33,35,38);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,20)(14,15)(16,22)(17,18)(19,21)(23,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,38)(33,39)(34,40);(1,2)(3,4)(5,7)(6,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,32)(27,29)(30,36)(31,33)(34,39)(35,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,20)(17,21)(18,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,11)(12,21)(14,17)(15,18)(16,24)(19,20)(22,23)(25,29,30,35)(26,33,34,38)(27,36,37,28)(31,39,40,32)Polycyclic presentation:
2^9|a^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625733);