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 |
| prime radical | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,12,16)(4,6,11,14)(5,13,15,10)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,39,40)(35,37,41,36)(38,42,47,48)(43,45,46,44);(1,3,7,12)(2,8,9,16)(4,10,11,13)(5,14,15,6)(19,27)(20,30)(23,32)(24,29)(34,40)(36,37)(42,48)(44,45);(2,9)(3,12)(6,14)(10,13)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31)(33,35)(34,36)(37,40)(38,46)(39,41)(42,45)(43,47)(44,48);(1,4,7,11)(2,6,9,14)(3,13,12,10)(5,8,15,16)(33,36,39,37)(34,41,40,35)(38,44,47,45)(42,46,48,43);(1,5)(2,10)(3,6)(4,8)(7,15)(9,13)(11,16)(12,14)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,39,36)(34,35,40,41)(38,45,47,44)(42,43,48,46);(1,6)(2,11)(3,5)(4,9)(7,14)(8,13)(10,16)(12,15)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109380);