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 | |
4123 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,21,24,26)(20,23,31,29)(27,30,28,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(2,6)(3,9)(8,13)(11,15)(17,19)(18,21)(20,28)(22,24)(23,32)(25,26)(27,31)(29,30)(35,39)(36,42)(43,45)(44,47);(4,12)(8,13)(11,15)(14,16)(19,24)(21,26)(27,28)(30,32)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,26)(20,29)(21,24)(22,25)(23,31)(27,30)(28,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8325572);