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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,26)(19,22,28,32)(20,25,21,24)(27,30,29,31)(33,34)(35,45)(36,42)(37,39)(38,43)(40,44)(41,47)(46,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,22)(20,29)(21,27)(23,28)(24,30)(25,31)(26,32)(33,35,40,47)(34,41,44,45)(36,46,38,37)(39,43,48,42);(2,6)(7,14)(8,11)(15,16)(17,20,23,21)(18,24,26,25)(19,27,28,29)(22,31,32,30)(33,36,40,38)(34,42,44,43)(35,46,47,37)(39,45,48,41);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,21,23,20)(18,25,26,24)(19,27,28,29)(22,31,32,30)(33,37)(34,39)(35,36)(38,47)(40,46)(41,43)(42,45)(44,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,19)(18,22)(20,27)(21,29)(23,28)(24,31)(25,30)(26,32)(33,38,40,36)(34,43,44,42)(35,46,47,37)(39,45,48,41);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,19)(20,30)(21,31)(23,32)(24,29)(25,27)(26,28)(33,39)(34,37)(35,43)(36,41)(38,45)(40,48)(42,47)(44,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,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, 8832567);