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 | #50784 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,9,5)(4,8,12,15)(11,16,13,14)(17,18,22,25)(19,26,29,23)(20,21,24,31)(27,28,30,32)(33,34)(35,40)(36,38)(37,42)(39,41)(43,47)(44,48)(45,46);(2,7)(3,9)(8,15)(11,13)(17,19,22,29)(18,23,25,26)(20,30,24,27)(21,28,31,32)(33,35,39,44)(34,40,41,48)(36,46,43,37)(38,45,47,42);(2,7)(4,12)(5,10)(11,13)(20,24)(21,31)(27,30)(28,32)(34,41)(35,44)(37,46)(38,47);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,26)(31,32)(33,36,39,43)(34,38,41,47)(35,37,44,46)(40,42,48,45);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,20,22,24)(18,21,25,31)(19,27,29,30)(23,32,26,28)(33,37)(34,42)(35,36)(38,40)(39,46)(41,45)(43,44)(47,48);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,21)(18,24)(19,28)(20,25)(22,31)(23,27)(26,30)(29,32)(33,38)(34,43)(35,45)(36,41)(37,40)(39,47)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947686);