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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,2,6,7)(3,4,10,9)(5,8,15,13)(11,12,16,14)(18,20)(23,24)(25,28)(30,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(3,10)(8,13)(11,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,20,22,18)(19,24,27,23)(21,25,29,28)(26,30,32,31)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,15)(7,13)(9,14)(10,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9088960);