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 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,20)(21,24)(22,30)(23,25)(26,31)(27,32)(28,29)(33,34)(35,39)(36,43)(37,46)(38,41)(40,45)(42,48)(44,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(33,35,38,37)(34,39,41,46)(36,44,45,42)(40,48,43,47);(2,9)(3,4)(10,16)(11,12)(18,25)(20,24)(26,30)(28,32)(33,36,38,45)(34,40,41,43)(35,42,37,44)(39,47,46,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,19)(18,24,25,20)(22,27,31,29)(26,28,30,32)(33,34)(35,39)(36,40)(37,46)(38,41)(42,47)(43,45)(44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37,38,35)(34,39,41,46)(36,44,45,42)(40,47,43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9845230);