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 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,39,36)(35,42,40,37)(38,44,46,41)(43,45,47,48);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(35,40)(37,42)(43,47)(45,48);(1,3,6,4)(2,8,10,7)(5,11,13,12)(9,15,16,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gh,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, 10329796);