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 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,12,7)(4,9,15,14)(5,16,13,10)(17,18)(19,26)(20,25)(21,29)(22,27)(23,28)(24,30)(31,32)(33,34,39,37)(35,40,41,36)(38,45,46,42)(43,44,48,47);(1,3,6,12)(2,7,8,11)(4,13,15,5)(9,16,14,10)(17,19)(18,24)(20,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,35)(34,40)(36,37)(38,43)(39,41)(42,47)(44,45)(46,48);(2,8)(7,11)(9,14)(10,16)(17,20,23,21)(18,25,28,29)(19,27,30,31)(22,24,32,26)(33,36,39,40)(34,41,37,35)(38,44,46,47)(42,48,45,43);(1,2)(3,11)(4,14)(5,10)(6,8)(7,12)(9,15)(13,16)(17,21,23,20)(18,25,28,29)(19,27,30,31)(22,26,32,24)(33,36)(34,35)(37,41)(38,44)(39,40)(42,43)(45,48)(46,47);(1,4,6,15)(2,9,8,14)(3,5,12,13)(7,10,11,16)(17,19)(18,26)(20,27)(21,31)(22,25)(23,30)(24,28)(29,32)(33,37,39,34)(35,40,41,36)(38,42,46,45)(43,44,48,47);(1,5)(2,10)(3,4)(6,13)(7,9)(8,16)(11,14)(12,15)(17,22)(18,27)(19,29)(20,24)(21,26)(23,32)(25,30)(28,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154523);