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 | #45074 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4007 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(2,7)(3,8)(9,16)(10,13)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,7)(3,8)(9,16)(10,13)(20,28)(25,30)(27,29)(31,32)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38,39,35)(34,42,43,40)(36,45,47,44)(37,46,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657409);