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 | |
4109 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)(3,7)(4,5)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18,23,27)(19,21,26,24)(20,22,30,25)(28,31,32,29)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19,23,26)(18,24,27,21)(20,28,30,32)(22,29,25,31)(35,37)(39,42)(43,45)(47,48);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,23,30)(18,25,27,22)(19,28,26,32)(21,29,24,31)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,20)(18,22)(19,28)(21,31)(23,30)(24,29)(25,27)(26,32)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,24)(25,32)(30,31)(33,37,38,35)(34,39,41,42)(36,43,46,45)(40,48,44,47);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,22)(18,20)(19,29)(21,32)(23,25)(24,28)(26,31)(27,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113970);