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 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(18,23)(19,24)(25,30)(26,31)(33,34)(35,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(2,7)(4,14)(8,11)(10,15)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,39,47)(34,40,38,46)(36,44,45,37)(41,43,42,48);(2,7)(3,12)(4,14)(5,16)(19,24)(20,27)(21,29)(25,30)(33,36,39,45)(34,41,38,42)(35,37,47,44)(40,48,46,43);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36)(34,42)(35,44)(37,47)(38,41)(39,45)(40,48)(43,46);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,44)(34,43,38,48)(35,45,47,36)(40,41,46,42);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,34)(35,46,47,40)(36,41,45,42)(37,48,44,43)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922530);