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 | |
3842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(37,45)(42,46)(43,44)(47,48);(3,9)(6,12)(10,15)(13,16)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32)(34,39)(40,41)(42,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038404);