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 | |
4184 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)(3,9,11,5)(4,12,13,8)(10,14,16,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(34,39)(36,40)(41,45)(44,47);(1,3,6,11)(2,5,7,9)(4,10,13,16)(8,14,12,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(2,7)(4,13)(5,9)(10,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,4,6,13)(2,8,7,12)(3,10,11,16)(5,14,9,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5)(2,3)(4,14)(6,9)(7,11)(8,10)(12,16)(13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,14)(12,15)(13,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669203);