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 | #49754 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,9,4,8)(5,14,15,10)(6,11,16,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,46)(45,48);(1,3,7,4)(2,8,12,9)(5,11,15,13)(6,10,16,14)(19,20)(21,29)(23,26)(24,31)(33,35,39,38)(34,40,42,43)(36,44,47,45)(37,48,41,46);(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36)(34,41)(35,44)(37,42)(38,45)(39,47)(40,46)(43,48);(1,4,7,3)(2,9,12,8)(5,11,15,13)(6,10,16,14)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36)(34,41)(35,45)(37,42)(38,44)(39,47)(40,48)(43,46);(1,5)(2,10)(3,13)(4,11)(6,8)(7,15)(9,16)(12,14)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,41)(34,36,42,47)(35,46,38,48)(40,45,43,44);(1,6,7,16)(2,11,12,13)(3,10,4,14)(5,8,15,9)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,38,39,35)(34,40,42,43)(36,45,47,44)(37,48,41,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,e^2=hi,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, 8400409);