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 | |
4150 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)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,21)(20,25)(22,26)(23,27)(24,30)(28,29)(31,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,30)(18,24,27,21)(20,28,26,32)(22,29,25,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(3,11)(4,12)(6,16)(8,13)(20,26)(22,25)(28,32)(29,31)(33,34,38,41)(35,36,39,42)(37,45,46,40)(43,48,47,44);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,23,26)(18,25,27,22)(19,28,30,32)(21,29,24,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,23,24)(18,19,27,30)(20,31,26,29)(22,28,25,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22,23,25)(18,26,27,20)(19,29,30,31)(21,28,24,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=gh,e^2=gh,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, 8835246);