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 | |
4135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,12)(14,16)(17,18)(19,25)(20,29)(21,24)(22,31)(23,26)(27,28)(30,32)(37,45)(40,47)(43,46)(44,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,23,21)(18,24,26,25)(20,27,31,30)(22,28,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,23,21)(18,25,26,24)(20,30,31,27)(22,28,29,32)(33,35,38,39)(34,36,41,42)(37,43,45,46)(40,44,47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,22)(19,27)(21,30)(23,31)(24,28)(25,32)(26,29)(33,36,38,42)(34,39,41,35)(37,44,45,48)(40,46,47,43);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21,23,19)(18,24,26,25)(20,27,31,30)(22,32,29,28)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8068468);