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 | #45529 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,23,26)(19,27,21,24)(20,22,31,25)(28,32,30,29)(33,34)(35,40)(36,44)(37,38)(39,43)(41,47)(42,48)(45,46);(3,9)(6,13)(10,15)(14,16)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32)(33,35)(34,40)(36,37)(38,44)(39,45)(41,48)(42,47)(43,46);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,31)(18,25,26,22)(19,28,21,30)(24,32,27,29)(33,36,39,47)(34,41,43,44)(35,42,45,37)(38,46,48,40);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,22)(19,28)(21,30)(23,31)(24,29)(25,26)(27,32)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,45)(43,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32)(33,37)(34,38)(35,36)(39,42)(40,44)(41,46)(43,48)(45,47);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,20)(19,29)(21,32)(23,25)(24,28)(26,31)(27,30)(33,38)(34,42)(35,44)(36,46)(37,43)(39,48)(40,47)(41,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817272);