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 | |
4156 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,8)(4,9)(7,14)(10,15)(18,24)(19,27)(20,29)(26,30)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073946);