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 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,6)(7,9)(8,12)(15,16)(18,25)(19,29)(20,27)(26,32)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,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, 10050975);