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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,27)(19,28,32,24)(20,25,30,22)(21,26,29,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,46)(44,48)(45,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,23,32)(18,24,27,28)(20,29,30,21)(22,31,25,26)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(2,6)(3,9)(4,12)(14,16)(20,30)(21,29)(22,25)(26,31)(34,38)(36,42)(37,45)(43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20)(18,25)(19,29)(21,32)(22,27)(23,30)(24,26)(28,31)(33,36,39,42)(34,40,38,35)(37,46,45,44)(41,43,47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,23,29)(18,26,27,31)(19,30,32,20)(22,24,25,28)(33,37,39,45)(34,41,38,47)(35,43,40,48)(36,44,42,46);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22,23,25)(18,20,27,30)(19,31,32,26)(21,24,29,28)(33,38,39,34)(35,42,40,36)(37,47,45,41)(43,46,48,44)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8964549);