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 | |
3844 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)(3,8,6,4)(5,9,15,16)(11,13,14,12)(17,18,24,28)(19,23,21,26)(20,30,32,25)(22,27,29,31)(34,40)(35,45)(36,38)(43,48);(1,3,7,6)(2,8,10,4)(5,11,15,14)(9,13,16,12)(17,19,24,21)(18,23,28,26)(20,22,32,29)(25,31,30,27)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(1,4,7,8)(2,3,10,6)(5,13,15,12)(9,14,16,11)(17,20,24,32)(18,25,28,30)(19,29,21,22)(23,27,26,31)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,4)(2,3)(5,13)(6,10)(7,8)(9,14)(11,16)(12,15)(17,21,24,19)(18,26,28,23)(20,29,32,22)(25,27,30,31)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22,24,29)(18,27,28,31)(19,20,21,32)(23,30,26,25)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,6,7,3)(2,4,10,8)(5,14,15,11)(9,12,16,13)(17,23,24,26)(18,19,28,21)(20,31,32,27)(22,30,29,25)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=i,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, 8936851);