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 | |
4109 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(2,4)(7,9)(8,12)(15,16)(18,19)(23,25)(24,26)(30,31)(35,44)(37,45)(40,48)(41,46);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400414);