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 | |
3996 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)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,20)(18,24,25,21)(22,28,31,27)(26,29,30,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(2,6)(7,10)(8,12)(14,16)(18,25)(19,20)(26,30)(27,28)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,21,25,24)(22,28,31,27)(26,29,30,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21)(18,19)(20,25)(22,29)(23,24)(26,27)(28,30)(31,32)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,31)(18,26,25,30)(19,27,20,28)(21,29,24,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=hi,f^2=gh,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, 8663433);