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 |
| prime radical | #2661 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,5,8,13)(4,6,9,15)(11,12,14,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(35,41)(38,42)(43,47)(46,48);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(18,24)(19,20)(26,29)(27,28)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(3,8)(5,13)(11,14)(12,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528674);