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 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,41)(36,40)(37,45)(38,42)(39,47)(43,46)(44,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35)(34,37)(36,43)(38,45)(39,46)(40,48)(41,42)(44,47);(3,11)(7,8)(12,16)(13,14)(20,28)(25,30)(27,29)(31,32)(33,36,38,47)(34,39,42,40)(35,43,45,44)(37,46,41,48);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36)(34,40)(35,44)(37,46)(38,47)(39,42)(41,48)(43,45);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37)(34,35)(36,46)(38,41)(39,43)(40,44)(42,45)(47,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,35)(34,41)(36,43)(37,42)(38,45)(39,48)(40,46)(44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8641177);