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 | |
4276 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)(5,12)(7,11)(8,14)(10,15)(13,16)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(2,4)(7,9)(8,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(35,37)(40,42)(45,47)(46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329563);