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 | |
3834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,13)(4,9,14,16)(5,12,15,10)(17,18,21,23)(19,24,27,22)(20,28,30,25)(26,31,29,32)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19,21,27)(18,22,23,24)(20,26,30,29)(25,31,28,32)(33,35)(34,41)(36,39)(37,45)(38,44)(40,46)(42,48)(43,47);(2,8)(3,11)(9,16)(10,12)(18,23)(22,24)(25,28)(31,32)(33,36,40,37)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,18)(19,22)(20,25)(21,23)(24,27)(26,31)(28,30)(29,32)(33,37,40,36)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,19)(18,24)(20,29)(21,27)(22,23)(25,31)(26,30)(28,32)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,20)(18,25)(19,26)(21,30)(22,31)(23,28)(24,32)(27,29)(33,39,40,45)(34,43,38,48)(35,36,46,37)(41,47,44,42)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,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, 10050931);