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 | |
4147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,13,14,9)(11,12,16,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34)(35,41)(36,38)(37,47)(39,42)(40,46)(43,48)(44,45);(1,3,6,10)(2,7,8,4)(5,11,14,16)(9,15,13,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(2,8)(3,10)(5,14)(12,15)(19,25)(21,29)(23,26)(24,28)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,45,41,48);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,45,41,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,15)(12,13)(14,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,35)(34,41)(36,44)(37,43)(38,45)(39,46)(40,42)(47,48);(1,5,6,14)(2,9,8,13)(3,11,10,16)(4,12,7,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,36)(35,45)(37,42)(39,47)(40,43)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870559);