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 | |
4119 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,24,26,21)(20,22,29,32)(27,28,31,30)(34,39)(35,42)(37,45)(44,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,26)(18,24,25,21)(20,27,29,31)(22,28,32,30)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,7)(6,13)(8,11)(14,15)(18,25)(19,26)(20,29)(28,30)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,30)(22,27)(23,24)(25,26)(28,29)(31,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9689909);