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 | C2×#127 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,6,4,11)(5,9,13,16)(10,15,12,14)(19,25)(23,26)(27,31)(30,32)(33,34,39,45)(35,46,47,40)(36,42,48,41)(37,43,44,38);(1,3)(2,6)(4,7)(5,12)(8,11)(9,14)(10,13)(15,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35)(34,40)(36,38)(37,41)(39,47)(42,44)(43,48)(45,46);(2,8)(3,4)(5,13)(14,15)(18,20)(23,26)(24,28)(30,32)(33,36,39,48)(34,41,45,42)(35,38,47,43)(37,46,44,40);(1,4,7,3)(2,6,8,11)(5,12,13,10)(9,15,16,14)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,42)(35,43)(37,40)(38,47)(39,48)(41,45)(44,46);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,6)(2,3)(4,8)(5,14)(7,11)(9,12)(10,16)(13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,43)(34,44,45,37)(35,36,47,48)(40,42,46,41)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976565);