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 | #35473 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4033 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)(3,6)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,38,40)(34,39,42,36)(37,43,46,45)(41,47,48,44);(2,3)(5,13)(7,9)(11,16)(19,27)(20,30)(23,31)(24,29)(33,35,38,40)(34,36,42,39)(37,45,46,43)(41,47,48,44);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,36)(34,40)(35,42)(37,44)(38,39)(41,45)(43,48)(46,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841387);