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 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,14,7)(10,16,13,15)(17,18)(19,26)(20,29)(21,24)(22,28)(23,25)(27,31)(30,32)(33,34,38,43)(35,42,45,39)(36,46,47,40)(37,41,44,48);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19)(18,22)(20,30)(21,27)(23,28)(24,32)(25,26)(29,31)(33,35)(34,39)(36,44)(37,47)(38,45)(40,48)(41,46)(42,43);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,23,21)(18,24,25,29)(19,27,28,30)(22,31,26,32)(33,36,38,47)(34,40,43,46)(35,44,45,37)(39,48,42,41);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,19)(18,22)(20,27)(21,30)(23,28)(24,31)(25,26)(29,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,46)(43,48)(45,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,21,23,20)(18,24,25,29)(19,27,28,30)(22,32,26,31)(33,35)(34,42)(36,44)(37,47)(38,45)(39,43)(40,41)(46,48);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,22)(18,19)(20,31)(21,32)(23,26)(24,27)(25,28)(29,30)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411766);