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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,7,4)(5,8,14,16)(11,15,13,12)(17,18,22,26)(19,27,28,23)(20,24,30,32)(21,29,31,25)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(1,3,6,7)(2,4,9,10)(5,13,14,11)(8,15,16,12)(19,28)(21,31)(23,27)(25,29)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,2,6,9)(3,4,7,10)(5,8,14,16)(11,12,13,15)(17,19)(18,23)(20,29)(21,24)(22,28)(25,30)(26,27)(31,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,32)(28,29)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,7)(3,9)(5,12)(6,10)(8,13)(11,16)(14,15)(17,19)(18,23)(20,29)(21,24)(22,28)(25,30)(26,27)(31,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(1,5,6,14)(2,8,9,16)(3,11,7,13)(4,12,10,15)(17,21)(18,25)(19,24)(20,27)(22,31)(23,30)(26,29)(28,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757876);