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 | |
3980 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)(3,9,4,5)(6,10,14,15)(11,16,12,13)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34)(35,40)(36,45)(37,41)(38,44)(39,42)(43,47)(46,48);(2,8)(5,9)(10,15)(13,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,35)(34,40)(36,46)(37,43)(38,42)(39,44)(41,47)(45,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36,39,37)(34,41,42,45)(35,43,44,46)(38,47,40,48);(1,4,7,3)(2,9,8,5)(6,11,14,12)(10,13,15,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,37,39,36)(34,41,42,45)(35,43,44,46)(38,48,40,47);(1,5,7,9)(2,4,8,3)(6,13,14,16)(10,12,15,11)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,43)(42,44)(45,46);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,35)(34,38)(36,43)(37,46)(39,44)(40,42)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327394);