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 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,27,21,24)(20,26,31,22)(28,32,30,29)(33,34,38,43)(35,41,44,39)(36,40,45,46)(37,42,47,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32)(33,35)(34,39)(36,42)(37,40)(38,44)(41,43)(45,48)(46,47);(3,10)(6,7)(11,16)(14,15)(18,25)(20,31)(24,27)(28,30)(33,36,38,45)(34,40,43,46)(35,42,44,48)(37,41,47,39);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,31)(18,26,25,22)(19,28,21,30)(24,29,27,32)(33,36)(34,40)(35,42)(37,39)(38,45)(41,47)(43,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,26,32)(33,35)(34,41)(36,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)(33,37,38,47)(34,42,43,48)(35,40,44,46)(36,41,45,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635745);