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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,15)(4,14,13,6)(5,9,12,16)(17,18,21,24)(19,22,26,30)(20,27,28,23)(25,31,32,29)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,11)(2,8,10,15)(4,12,13,5)(6,9,14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(3,11)(4,13)(6,14)(8,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4)(2,6)(3,12)(5,11)(7,13)(8,9)(10,14)(15,16)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,12)(2,9,10,16)(3,13,11,4)(6,15,14,8)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,4)(3,9)(5,15)(7,14)(8,12)(10,13)(11,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839558);