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 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,44)(35,42,37,40)(36,43,38,41)(45,47,46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(2,6)(7,10)(8,12)(14,16)(18,22)(23,25)(24,28)(30,31)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663232);