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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,34)(35,40)(36,39)(37,45)(38,42)(41,46)(43,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,31)(29,30)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,6)(7,12)(8,13)(14,16)(18,21)(19,25)(24,29)(26,28)(33,35,38,36)(34,40,42,39)(37,43,46,44)(41,48,45,47);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20,23,31)(18,24,21,29)(19,26,25,28)(22,30,27,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,21,23,18)(19,27,25,22)(20,29,31,24)(26,32,28,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,22,23,27)(18,25,21,19)(20,30,31,32)(24,28,29,26)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025847);