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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34,38,43)(35,41,45,39)(36,44,46,40)(37,42,48,47);(2,7)(4,9)(8,13)(12,15)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,35)(34,39)(36,42)(37,44)(38,45)(40,48)(41,43)(46,47);(2,7)(3,10)(5,14)(12,15)(18,25)(19,20)(22,31)(29,32)(36,46)(37,48)(40,44)(42,47);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32)(33,36)(34,40)(35,42)(37,41)(38,46)(39,48)(43,44)(45,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,20)(19,25)(22,29)(23,24)(26,28)(27,30)(31,32)(33,35)(34,41)(36,47)(37,44)(38,45)(39,43)(40,48)(42,46);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37,38,48)(34,42,43,47)(35,44,45,40)(36,41,46,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815649);