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 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,6)(4,5,10,9)(13,15,16,14)(17,18)(19,21)(20,24)(22,26)(23,25)(27,29)(28,30)(31,32)(33,34)(35,40)(36,46)(37,41)(38,47)(39,42)(43,44)(45,48);(1,3,7,12)(2,8,11,6)(4,13,10,16)(5,15,9,14)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(33,35,39,45)(34,40,42,48)(36,41,47,43)(37,38,44,46);(3,12)(4,10)(5,9)(6,8)(17,18,22,26)(19,23,25,21)(20,28,30,24)(27,29,32,31)(33,36,39,47)(34,38,42,46)(35,43,45,41)(37,48,44,40);(1,4)(2,9)(3,13)(5,11)(6,15)(7,10)(8,14)(12,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(33,36)(34,38)(35,41)(37,48)(39,47)(40,44)(42,46)(43,45);(1,5,7,9)(2,10,11,4)(3,14,12,15)(6,16,8,13)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,37,39,44)(34,41,42,43)(35,38,45,46)(36,40,47,48);(1,6)(2,3)(4,15)(5,16)(7,8)(9,13)(10,14)(11,12)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(33,38)(34,36)(35,44)(37,45)(39,46)(40,41)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=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, 8961676);