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 | #48279 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,32)(29,31)(33,34,38,43)(35,39,45,47)(36,41,46,40)(37,44,48,42);(1,2,6,7)(3,8,10,15)(4,5,9,14)(11,16,13,12)(17,19,23,28)(18,24,20,27)(21,30,31,26)(22,25,32,29)(33,35,38,45)(34,39,43,47)(36,42,46,44)(37,40,48,41);(2,7)(3,10)(5,14)(11,13)(18,20)(22,32)(24,27)(25,29)(33,36,38,46)(34,40,43,41)(35,44,45,42)(37,47,48,39);(1,3)(2,8)(4,13)(5,12)(6,10)(7,15)(9,11)(14,16)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26)(33,36)(34,41)(35,42)(37,39)(38,46)(40,43)(44,45)(47,48);(1,4,6,9)(2,5,7,14)(3,11,10,13)(8,16,15,12)(17,21,23,31)(18,25,20,29)(19,26,28,30)(22,27,32,24)(33,35)(34,39)(36,44)(37,40)(38,45)(41,48)(42,46)(43,47);(1,5)(2,9)(3,12)(4,7)(6,14)(8,11)(10,16)(13,15)(17,22,23,32)(18,26,20,30)(19,25,28,29)(21,27,31,24)(33,37,38,48)(34,42,43,44)(35,40,45,41)(36,39,46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8826506);