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 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,14)(4,9,13,16)(5,12,15,10)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47);(2,7)(4,13)(8,14)(10,12)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,38)(36,45)(37,46)(39,42)(40,41)(43,44)(47,48);(2,7)(3,11)(9,16)(10,12)(19,26)(22,23)(27,32)(30,31)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,41,44,48);(1,3)(2,8)(4,10)(5,9)(6,11)(7,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,45)(37,42)(38,41)(39,46)(43,47)(44,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,8,15,14)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,40,44,47);(1,5,6,15)(2,10,7,12)(3,9,11,16)(4,14,13,8)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,38,39,44)(34,42,43,35)(36,41,46,48)(37,47,45,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=g,f^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, 9074001);