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 | |
4224 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)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,3)(2,4)(5,11)(6,7)(8,10)(9,12)(13,16)(14,15)(17,19,23,31)(18,24,22,30)(20,28,29,21)(25,27,26,32)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(1,4,6,8)(2,7,10,3)(5,13,14,12)(9,11,16,15)(17,20,23,29)(18,25,22,26)(19,21,31,28)(24,32,30,27)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,4)(2,3)(5,12)(6,8)(7,10)(9,11)(13,14)(15,16)(17,20)(18,26)(19,28)(21,31)(22,25)(23,29)(24,32)(27,30)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,3)(2,8)(4,10)(5,11)(6,7)(9,13)(12,16)(14,15)(17,21,23,28)(18,27,22,32)(19,29,31,20)(24,25,30,26)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,5,6,14)(2,9,10,16)(3,11,7,15)(4,12,8,13)(17,22,23,18)(19,30,31,24)(20,25,29,26)(21,32,28,27)(33,37)(34,41)(35,44)(36,42)(38,46)(39,47)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922564);