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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,11)(3,12,7,9)(4,6,5,10)(13,15,14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,40)(35,37)(42,47)(43,45);(1,3,8,7)(2,9,11,12)(4,14,5,13)(6,15,10,16)(19,25)(22,28)(26,31)(29,32)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,5,8,4)(2,6,11,10)(3,14,7,13)(9,16,12,15)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,6)(2,4)(3,15)(5,11)(7,16)(8,10)(9,14)(12,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,16,10,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,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, 8324304);