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 |
| prime radical | #55601 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,23,22)(19,27,30,24)(20,25,29,31)(21,26,28,32)(33,34,39,43)(35,44,37,40)(36,41,42,38)(45,48,47,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,26)(31,32)(33,34)(35,40)(36,41)(37,44)(38,42)(39,43)(45,46)(47,48);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,23,29)(18,25,22,31)(19,28,30,21)(24,26,27,32)(33,35,39,37)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,26)(27,32)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,46,44,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,25,27,31)(33,37,39,35)(34,40,43,44)(36,45,42,47)(38,46,41,48);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,26,28,32)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,45,44,47)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=h,e^2=gi,f^2=g,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, 9113957);