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 | |
4216 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)(3,10,9,5)(4,12,13,8)(11,14,15,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,40)(36,45)(37,47)(38,43)(39,44)(41,46)(42,48);(2,7)(3,9)(4,13)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(35,44)(36,46)(39,40)(41,45);(2,7)(3,9)(8,12)(11,15)(18,22)(23,25)(24,29)(30,31)(33,35,38,44)(34,39,43,40)(36,42,46,47)(37,41,48,45);(1,3,6,9)(2,5,7,10)(4,11,13,15)(8,14,12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,40)(36,42)(37,41)(38,44)(39,43)(45,48)(46,47);(1,4,6,13)(2,8,7,12)(3,11,9,15)(5,14,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,41)(35,42)(37,40)(38,46)(39,48)(43,45)(44,47);(1,5)(2,9)(3,7)(4,14)(6,10)(8,15)(11,12)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,42)(35,41)(36,40)(38,48)(39,46)(43,47)(44,45)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017926);