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 | |
4276 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)(3,8)(4,13)(5,9)(6,7)(10,11)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,7)(4,9)(8,15)(10,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(5,13)(8,15)(11,16)(20,28)(23,30)(26,32)(29,31)(34,38)(39,41)(40,44)(47,48);(1,3,6,12)(2,8,7,15)(4,14,9,10)(5,11,13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,6,9)(2,5,7,13)(3,10,12,14)(8,16,15,11)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5)(2,9)(3,11)(4,7)(6,13)(8,10)(12,16)(14,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|b^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313996);