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 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,14,9)(4,15,5,6)(11,13,12,16)(17,18)(19,27)(20,31)(21,32)(22,26)(23,24)(25,28)(29,30)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,7,14)(2,8,10,9)(4,12,5,11)(6,13,15,16)(18,24)(19,28)(20,32)(26,30)(35,43)(36,46)(39,48)(40,45);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,16,9,13)(17,19,23,28)(18,25,24,27)(20,29,32,22)(21,30,31,26)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,16,9,13)(17,20)(18,21)(19,29)(22,28)(23,32)(24,31)(25,30)(26,27);(1,3)(2,9)(4,11)(5,12)(6,13)(7,14)(8,10)(15,16)(17,21)(18,20)(19,30)(22,27)(23,31)(24,32)(25,29)(26,28)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,6)(2,4)(3,13)(5,10)(7,15)(8,12)(9,11)(14,16)(17,22,23,29)(18,26,24,30)(19,20,28,32)(21,27,31,25)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109660);