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 | |
4305 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)(3,8)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,23)(19,20,25,24)(21,29,30,26)(27,31,32,28)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(2,7)(4,14)(8,16)(10,12)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,31)(30,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(3,11)(5,15)(8,16)(10,12)(18,23)(19,25)(21,30)(28,31)(34,39)(36,40)(37,46)(42,48);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,48)(46,47);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837463);