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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,28)(20,31)(21,22)(23,25)(24,30)(26,29)(27,32)(33,34,39,43)(35,40,44,46)(36,38,42,48)(37,41,47,45);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,30)(18,24,25,28)(20,22,29,32)(21,31,27,26)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,46)(47,48);(4,11)(7,9)(12,16)(14,15)(18,25)(21,27)(24,28)(26,31)(35,44)(37,47)(40,46)(41,45);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20)(18,26)(19,22)(21,24)(23,29)(25,31)(27,28)(30,32)(33,36)(34,38)(35,45)(37,40)(39,42)(41,44)(43,48)(46,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,27)(18,22,25,32)(19,26,30,31)(20,28,29,24)(33,37)(34,41)(35,38)(36,46)(39,47)(40,42)(43,45)(44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,27)(19,29)(20,30)(21,25)(23,32)(24,26)(28,31)(33,38)(34,42)(35,37)(36,43)(39,48)(40,41)(44,47)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973308);