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 | |
4291 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(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)(7,9)(8,12)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(36,40)(37,46)(43,48);(3,10)(7,9)(11,13)(15,16)(18,22)(23,26)(24,28)(30,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,45)(46,48);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(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)(11,15)(12,14)(13,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8782757);