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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,23)(20,26)(21,24)(22,30)(25,28)(27,29)(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)(17,19)(18,25)(20,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(2,6)(7,10)(8,12)(14,16)(17,20,24,22)(18,26,21,30)(19,27,28,31)(23,29,25,32)(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)(17,21,24,18)(19,23,28,25)(20,26,22,30)(27,32,31,29)(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)(17,22,24,20)(18,26,21,30)(19,27,28,31)(23,32,25,29)(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)(17,23)(18,19)(20,29)(21,28)(22,32)(24,25)(26,27)(30,31)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,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, 8663291);