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 | |
4224 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)(3,10)(4,9)(5,14)(6,7)(8,13)(11,16)(12,15)(33,34,40,42)(35,36,38,37)(39,47,48,41)(43,45,46,44);(2,7)(4,14)(8,10)(11,15)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,47)(46,48);(2,7)(5,9)(8,10)(12,16)(18,23)(21,31)(24,32)(25,28)(33,36,40,37)(34,35,42,38)(39,45,48,44)(41,46,47,43);(1,3,6,13)(2,8,7,10)(4,15,14,11)(5,12,9,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,31)(27,30)(33,37,40,36)(34,35,42,38)(39,45,48,44)(41,43,47,46);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,16)(13,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,38,40,35)(34,37,42,36)(39,46,48,43)(41,45,47,44);(1,5,6,9)(2,4,7,14)(3,12,13,16)(8,15,10,11)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,39,40,48)(34,41,42,47)(35,43,38,46)(36,44,37,45)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805238);