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 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,10)(11,15)(14,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,16)(14,15)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(34,37)(35,40)(41,45)(43,47);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,3)(2,6)(4,11)(5,9)(7,14)(8,10)(12,16)(13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^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, 8738379);