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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,39)(35,42)(41,47)(43,45);(1,3,6,8)(2,7,10,4)(5,11,13,15)(9,14,16,12)(18,19)(23,25)(24,26)(30,31)(33,34,38,39)(35,40,42,36)(37,41,46,47)(43,48,45,44);(5,13)(9,16)(11,15)(12,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,7)(9,15)(11,16)(13,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,40)(36,42)(37,41)(38,39)(43,48)(44,45)(46,47);(1,5,6,13)(2,9,10,16)(3,11,8,15)(4,12,7,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,41,39,47)(35,43,42,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324408);