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 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,11,12)(5,9,15,13)(6,14,16,10)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,38,41)(35,42,36,39)(37,40,46,45)(43,48,44,47);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,16)(10,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,4)(3,11)(5,15)(6,16)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,34,38,41)(35,39,36,42)(37,45,46,40)(43,48,44,47);(1,4,7,2)(3,12,11,8)(5,9,15,13)(6,14,16,10)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,5)(2,9)(3,10)(4,13)(6,8)(7,15)(11,14)(12,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,6,7,16)(2,10,4,14)(3,13,11,9)(5,8,15,12)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8934151);