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 | |
4099 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)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,6)(4,12)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(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)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,f^2=g,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, 8659054);