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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,4,5,9)(6,14,15,10)(11,16,13,12)(19,25)(22,28)(26,31)(29,32)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(2,8)(4,9)(10,14)(12,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,40)(35,42)(41,44)(43,48);(1,3,7,5)(2,9,8,4)(6,11,15,13)(10,16,14,12)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,4,7,9)(2,3,8,5)(6,12,15,16)(10,11,14,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,3)(2,4,8,9)(6,13,15,11)(10,12,14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066000);