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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(19,25)(23,26)(27,31)(30,32)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(2,6)(4,12)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(34,39)(35,41)(42,45)(43,48);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,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, 8663491);