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 | |
4210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,12,14,8)(10,15,13,16)(19,25)(22,26)(27,31)(30,32)(33,34)(35,41)(36,39)(37,44)(38,42)(40,46)(43,47)(45,48);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,35,38,44)(34,37,42,41)(36,43,46,48)(39,45,40,47);(2,7)(6,9)(8,12)(15,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,36,38,46)(34,39,42,40)(35,43,44,48)(37,45,41,47);(1,2)(3,6)(4,12)(5,7)(8,14)(9,11)(10,16)(13,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,42)(41,45)(44,48);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,37,38,41)(34,35,42,44)(36,45,46,47)(39,43,40,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,35)(34,41)(36,43)(37,42)(38,44)(39,47)(40,45)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736419);