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 | |
4191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,14,8)(10,15,13,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,34,39,43)(35,40,44,46)(36,38,42,48)(37,41,47,45);(2,4)(7,9)(8,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,46)(47,48);(2,4)(7,9)(8,12)(15,16)(19,25)(22,26)(27,31)(30,32)(35,44)(37,47)(40,46)(41,45);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,36)(34,38)(35,45)(37,40)(39,42)(41,44)(43,48)(46,47);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37)(34,41)(35,38)(36,46)(39,47)(40,42)(43,45)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,38)(34,42)(35,37)(36,43)(39,48)(40,41)(44,47)(45,46)Polycyclic presentation:
2^9|a^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973299);