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 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,32)(28,31)(29,30)(33,34,38,40)(35,36,42,44)(37,39,48,47)(41,43,45,46);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,35)(34,36)(37,45)(38,42)(39,46)(40,44)(41,48)(43,47);(3,11)(7,8)(12,14)(13,16)(20,28)(23,31)(26,29)(30,32)(33,36,38,44)(34,35,40,42)(37,43,48,46)(39,41,47,45);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,21,28)(18,23,25,31)(19,26,27,29)(22,30,24,32)(33,36)(34,35)(37,46)(38,44)(39,45)(40,42)(41,47)(43,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,19)(18,24)(20,29)(21,27)(22,25)(23,30)(26,28)(31,32)(33,37)(34,39)(35,41)(36,43)(38,48)(40,47)(42,45)(44,46);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,31)(29,32)(33,34)(35,36)(37,47)(38,40)(39,48)(41,46)(42,44)(43,45)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9077862);