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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3776 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)(8,15)(10,13)(11,16)(18,21)(23,26)(24,29)(30,32)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(20,28)(24,29)(25,31)(30,32)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,7)(5,12)(8,15)(11,16)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,32)(27,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833693);