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)(17,18,22,24)(19,25,27,21)(20,28,29,23)(26,30,32,31)(33,34)(35,40)(36,42)(37,46)(38,48)(39,44)(41,47)(43,45);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,30,28,31)(33,35,39,46)(34,40,44,37)(36,38,47,45)(41,43,42,48);(2,6)(7,10)(8,12)(14,16)(20,29)(23,28)(26,32)(30,31)(33,36,39,47)(34,41,44,42)(35,45,46,38)(37,43,40,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,23,24,28)(19,26,27,32)(21,30,25,31)(33,36)(34,42)(35,38)(37,43)(39,47)(40,48)(41,44)(45,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,21)(20,26)(22,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,35)(36,43)(38,42)(39,40)(41,45)(44,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)(33,38)(34,43)(35,36)(37,42)(39,45)(40,41)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,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, 8659131);