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 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,9)(6,12,14,10)(11,16,13,15)(33,34,40,44)(35,42,37,38)(36,41,39,43)(45,46,47,48);(1,3)(2,8)(4,9)(5,7)(6,13)(10,16)(11,14)(12,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,40,37)(34,38,44,42)(36,45,39,47)(41,48,43,46);(2,4)(3,5)(10,12)(11,13)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,36,40,39)(34,41,44,43)(35,45,37,47)(38,48,42,46);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,40,35)(34,42,44,38)(36,45,39,47)(41,48,43,46);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,38,40,42)(34,35,44,37)(36,46,39,48)(41,47,43,45);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,39,40,36)(34,43,44,41)(35,45,37,47)(38,48,42,46)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725349);