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)(3,8)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34)(35,42)(36,39)(37,45)(38,40)(41,46)(43,47)(44,48);(2,7)(4,14)(8,16)(10,12)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,35,38,36)(34,39,40,42)(37,43,46,44)(41,47,45,48);(3,11)(5,15)(8,16)(10,12)(18,25)(19,20)(22,31)(29,32)(34,40)(37,46)(39,42)(43,44);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,20,23,19)(18,24,25,21)(22,28,31,27)(26,32,30,29)(33,34)(35,39)(36,42)(37,41)(38,40)(43,47)(44,48)(45,46);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,21,23,24)(18,20,25,19)(22,29,31,32)(26,28,30,27)(33,36,38,35)(34,39,40,42)(37,44,46,43)(41,47,45,48);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37,38,46)(34,41,40,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838275);