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 | |
3973 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)(6,10)(11,15)(14,16)(17,18,23,28)(19,24,31,30)(20,27,22,25)(21,26,29,32)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(17,19)(18,24)(20,21)(22,29)(23,31)(25,32)(26,27)(28,30)(34,39)(36,41)(40,45)(44,47);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,20,23,22)(18,25,28,27)(19,29,31,21)(24,26,30,32)(33,34)(35,36)(37,40)(38,39)(41,43)(42,44)(45,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,29)(18,26,28,32)(19,20,31,22)(24,27,30,25)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,22,23,20)(18,27,28,25)(19,29,31,21)(24,26,30,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,31)(29,32)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=gh,e^2=gi,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, 8664190);