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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3811 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)(3,13)(4,10)(5,6)(7,12)(8,15)(9,16)(11,14)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(1,3)(2,8)(4,5)(6,9)(7,15)(10,11)(12,13)(14,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(1,4,7,16)(2,9,12,10)(3,5,15,14)(6,13,11,8)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,4)(2,10)(3,14)(5,15)(6,8)(7,16)(9,12)(11,13)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,5,7,14)(2,11,12,6)(3,4,15,16)(8,10,13,9)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,6,7,11)(2,5,12,14)(3,9,15,10)(4,13,16,8)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gh,e^2=h,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, 8657383);