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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,24)(19,21,27,30)(20,25,28,32)(22,29,31,26)(34,38)(35,41)(36,43)(46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19)(18,21)(20,22)(23,27)(24,30)(25,29)(26,32)(28,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(2,6)(7,10)(8,12)(14,16)(18,24)(19,27)(20,28)(26,29)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,28)(18,25,24,32)(19,22,27,31)(21,29,30,26)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,19)(20,29)(22,25)(23,30)(24,27)(26,28)(31,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,26)(19,28)(20,27)(21,25)(23,31)(24,29)(30,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659201);