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 | |
4216 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,23,25)(19,24,29,32)(20,26,30,31)(21,27,28,22);(2,7)(4,10)(9,13)(12,16)(17,19,23,29)(18,24,25,32)(20,28,30,21)(22,31,27,26)(33,34,39,42)(35,43,37,40)(36,38,41,48)(44,47,46,45);(2,7)(3,8)(5,14)(12,16)(18,25)(19,29)(20,30)(22,27)(33,35,39,37)(34,40,42,43)(36,46,41,44)(38,47,48,45);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,45,43,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,21,23,28)(18,27,25,22)(19,30,29,20)(24,31,32,26)(33,37,39,35)(34,40,42,43)(36,44,41,46)(38,47,48,45);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,22,23,27)(18,28,25,21)(19,31,29,26)(20,24,30,32)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,46)(43,44)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831848);