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 | |
4313 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,16)(12,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(34,39)(35,41)(37,45)(44,47);(2,7)(4,14)(8,10)(11,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,34,38,39)(35,36,41,40)(37,42,45,46)(43,44,48,47);(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,3)(2,8)(4,15)(5,16)(6,13)(7,10)(9,12)(11,14)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35)(34,36)(37,43)(38,41)(39,40)(42,44)(45,48)(46,47);(1,4)(2,9)(3,11)(5,7)(6,14)(8,16)(10,12)(13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36)(34,41)(35,39)(37,44)(38,40)(42,48)(43,46)(45,47);(1,5,6,9)(2,4,7,14)(3,12,13,16)(8,11,10,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37,38,45)(34,42,39,46)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319263);