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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,9)(5,15,16,6)(11,12,13,14)(17,18)(19,25)(20,29)(21,30)(22,24)(23,28)(26,32)(27,31)(33,34)(35,36)(37,40)(38,46)(39,42)(41,48)(43,45)(44,47);(1,3,7,4)(2,8,10,9)(5,11,16,13)(6,14,15,12)(17,19,22,28)(18,23,24,25)(20,26,30,31)(21,32,29,27)(33,35,39,40)(34,36,42,37)(38,47,48,43)(41,45,46,44);(3,4)(8,9)(11,13)(12,14)(18,24)(20,30)(23,25)(26,31)(33,36,39,37)(34,40,42,35)(38,44,48,45)(41,47,46,43);(1,4,7,3)(2,9,10,8)(5,11,16,13)(6,14,15,12)(17,20)(18,21)(19,26)(22,30)(23,32)(24,29)(25,27)(28,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,14)(9,12)(10,15)(17,19)(18,23)(20,31)(21,27)(22,28)(24,25)(26,30)(29,32)(33,37,39,36)(34,35,42,40)(38,45,48,44)(41,43,46,47);(1,6)(2,5)(3,12)(4,14)(7,15)(8,13)(9,11)(10,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,48)(40,47)(42,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8751788);