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 | |
4231 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)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2,5,8)(3,6,10,9)(4,7,12,14)(11,13,16,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(36,44)(39,46)(42,48)(45,47);(3,10)(4,12)(6,9)(7,14)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,5,10)(2,6,8,9)(4,11,12,16)(7,13,14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=g,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, 8669215);