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 | |
4287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,38,42)(35,41,45,39)(36,37,47,40)(43,48,46,44);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,38,45)(34,39,42,41)(36,43,47,46)(37,44,40,48);(2,5)(3,6)(4,11)(15,16)(19,26)(20,24)(21,29)(23,30)(33,36,38,47)(34,40,42,37)(35,43,45,46)(39,48,41,44);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36)(34,40)(35,43)(37,42)(38,47)(39,48)(41,44)(45,46);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,35)(34,41)(36,46)(37,44)(38,45)(39,42)(40,48)(43,47);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,37,38,40)(34,36,42,47)(35,44,45,48)(39,43,41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,e^2=gh,f^2=h,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, 8882585);