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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18)(19,28)(20,30)(21,27)(22,24)(23,25)(26,31)(29,32)(33,34,40,44)(35,41,38,43)(36,42,37,39)(45,46,47,48);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(33,35,40,38)(34,41,44,43)(36,47,37,45)(39,46,42,48);(2,9)(3,11)(4,13)(10,14)(18,25)(20,31)(24,28)(27,29)(33,36,40,37)(34,42,44,39)(35,45,38,47)(41,46,43,48);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,37,40,36)(34,39,44,42)(35,45,38,47)(41,46,43,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,38,40,35)(34,43,44,41)(36,47,37,45)(39,46,42,48);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,39)(34,36)(35,46)(37,44)(38,48)(40,42)(41,47)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10090344);