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 | |
4243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,13,16)(5,14,15,10)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,7)(4,13)(8,11)(10,14)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,7)(3,12)(9,16)(10,14)(18,25)(19,30)(20,27)(28,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,13)(15,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4)(2,9)(3,10)(5,11)(6,13)(7,16)(8,15)(12,14)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,15)(2,10,7,14)(3,9,12,16)(4,8,13,11)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073839);