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×#127 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,7,10,8)(4,9,12,13)(11,14,16,15)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32)(33,34)(35,43)(36,42)(37,41)(38,44)(39,40)(45,48)(46,47);(1,3)(2,7)(4,11)(5,8)(6,10)(9,14)(12,16)(13,15)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32)(34,40)(36,46)(41,43)(44,45);(3,10)(7,8)(11,16)(14,15)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30)(33,35,39,37)(34,41,40,43)(36,44,46,45)(38,42,48,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,20)(18,25)(19,29)(21,31)(22,27)(23,26)(24,32)(28,30)(33,36,39,46)(34,42,40,47)(35,45,37,44)(38,43,48,41);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,21,23,19)(18,24,27,28)(20,31,26,29)(22,30,25,32)(33,37,39,35)(34,43,40,41)(36,44,46,45)(38,42,48,47);(1,5,6,2)(3,8,10,7)(4,9,12,13)(11,14,16,15)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,41,46,43)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=g,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, 8654140);