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 | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,16)(4,8,14,13)(5,9,15,12)(17,18,23,26)(19,24,30,25)(20,31,21,22)(27,29,28,32)(33,34)(35,42)(36,44)(37,40)(38,39)(41,45)(43,47)(46,48);(3,11)(5,15)(7,16)(9,12)(17,19,23,30)(18,24,26,25)(20,28,21,27)(22,29,31,32)(33,35,39,37)(34,40,38,42)(36,43,45,46)(41,47,44,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,20,23,21)(18,22,26,31)(19,27,30,28)(24,32,25,29)(33,36)(34,41)(35,43)(37,46)(38,44)(39,45)(40,47)(42,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,21,23,20)(18,22,26,31)(19,28,30,27)(24,32,25,29)(33,34,39,38)(35,42,37,40)(36,41,45,44)(43,48,46,47);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,22)(18,20)(19,29)(21,26)(23,31)(24,28)(25,27)(30,32)(33,37,39,35)(34,40,38,42)(36,43,45,46)(41,48,44,47);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,30)(24,26)(31,32)(33,38,39,34)(35,40,37,42)(36,44,45,41)(43,47,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232527);