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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,11)(15,16)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,2,5,6)(3,7,9,13)(4,8,11,14)(10,15,12,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,12)(14,16)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,45,47);(1,3)(2,7)(4,12)(5,9)(6,13)(8,16)(10,11)(14,15)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,14)(12,16)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,4,5,11)(2,8,6,14)(3,10,9,12)(7,15,13,16)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319510);